====================================================================
BI-LIPSCHITZ ABSORPTION OF c_0
WITHOUT A LINEAR COPY OF c_0
====================================================================
A complete, self-contained rewrite for curious readers (from high
school upward). Every concept is built from scratch, every proof
appears in full, and every technical step is surrounded by intuition.
Original reference: "Bi-Lipschitz Absorption of c_0 Without a Linear
Copy of c_0" (OpenAI, September 2026).
CONVENTIONS AND NOTATION
------------------------
N = positive integers {1, 2, 3, ...}
R = reals
sup, inf = supremum, infimum
a_+ = max(a, 0) for a real number a
|c| = absolute value
||x|| = norm of x; a subscript indicates which space
c_0 = sequences (x_1, x_2, ...) with x_n -> 0, norm sup_n |x_n|
c_0(A) = same with index set A: f: A -> R with {a: |f(a)| > eps}
finite for every eps > 0, norm sup over A
l_inf = bounded real sequences, sup norm
l_2 = square-summable sequences, norm (sum |x_n|^2)^{1/2}
X (+) Y = direct sum with sup norm ||(x,y)|| = max(||x||,||y||)
B_X = closed unit ball of X
B = closed unit ball of D (always B_D here)
Lip(f) = least L >= 0 with ||f(x)-f(y)|| <= L ||x-y|| for all x,y
supp_B f = closure in B of { s in B : f(s) != 0 }
B_inf(p,r) = { y : ||y - p||_inf <= r }
dist_inf(p, K) = inf_{y in K} ||y - p||_inf
pi_j = the map D -> R^j keeping the first j coordinates of D
zeta = map B -> Z, zeta(s) = [delta_s - delta_0]
delta_s = the evaluation "molecule" at s (see Section 6)
All Banach spaces are real. Fractions like 1/2 are exact. Powers
like 2^{-j} are ordinary. Brackets [ ] are used only as ordinary
grouping, as interval notation, or as quotient notation; context
always makes the usage clear.
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PROLOGUE: WHAT THIS PAPER IS ABOUT
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1. BANACH SPACES IN ONE PARAGRAPH
A real vector space is a set where you can add elements and scale
by real numbers, with the usual rules (0 exists, negatives exist,
+ and * distribute, etc).
A norm ||.|| makes a vector space into a metric space: the distance
between x and y is ||x - y||. A norm satisfies
||x|| = 0 iff x = 0, ||c x|| = |c| ||x||,
||x + y|| <= ||x|| + ||y||.
A Banach space is a normed space that is complete: every Cauchy
sequence has a limit inside the space. Completeness is the
mathematician's way of saying "the space has no holes".
Two model Banach spaces:
c_0 = { (x_1, x_2, ...) : x_n -> 0 }, ||x|| = sup_n |x_n|.
l_2 = { (x_1, x_2, ...) : sum |x_n|^2 < +inf },
||x|| = (sum |x_n|^2)^{1/2}.
c_0 is the space of sequences that eventually get and stay small.
l_2 is the space of sequences whose squares are summable. They are
both infinite-dimensional, but with very different geometry: c_0
is "flat and cornered" (its unit ball has corners), whereas l_2 has
a round unit ball.
Throughout this paper, X, Y, Z, E, U, D denote real Banach spaces.
--------------------------------------------------------------------
2. LINEAR MAPS AND ISOMORPHISMS
A map T: X -> Y is linear if T(a x + b y) = a T(x) + b T(y) for
all x, y in X and a, b in R.
It is bounded if ||T(x)|| <= C ||x|| for some C. For linear maps
between Banach spaces, bounded = continuous.
A linear isomorphism is a bounded linear bijection whose inverse
is also bounded. Two Banach spaces are linearly isomorphic (or
"linearly the same") if such a map exists. Linear isomorphism
preserves the whole linear structure: subspaces, quotients,
bases, linear copies of other spaces.
An important special case: "X contains a linear copy of c_0"
means there is a closed linear subspace of X linearly isomorphic
to c_0. This is a strong linear property. If Z (+) c_0 and Z
were linearly isomorphic, then Z would contain a linear copy of
c_0, because {0} (+) c_0 is a linear copy of c_0 inside Z (+)
c_0.
--------------------------------------------------------------------
3. BI-LIPSCHITZ MAPS
A map F: X -> Y (not assumed linear) is Lipschitz if
||F(x) - F(y)|| <= L ||x - y|| (x, y in X)
for some constant L < +inf. The smallest such L is Lip(F), the
Lipschitz constant of F.
F is bi-Lipschitz if there are 0 < c <= C < +inf with
c ||x - y|| <= ||F(x) - F(y)|| <= C ||x - y|| (all x, y).
A bi-Lipschitz map is injective (if F(x) = F(y), the lower bound
forces x = y). If it is also surjective, it is a bi-Lipschitz
bijection, or a bi-Lipschitz equivalence.
Intuition: a bi-Lipschitz map is a coordinate change that can
distort distances by at most a fixed factor. It can twist, shear
and bend, but it cannot collapse distinct points and it cannot
suddenly push nearby points to be far apart.
--------------------------------------------------------------------
4. THE TWO NOTIONS COMPARED
Every bounded linear isomorphism is bi-Lipschitz (upper bound is
||T||, a lower bound is any c with 0 < c <= 1/||T^{-1}||). So
linearly isomorphic ==> bi-Lipschitz equivalent.
The converse is false, and the gap between the two is the subject
of this paper.
In finite dimensions, the gap is small: two finite-dimensional
normed spaces are bi-Lipschitz equivalent if and only if they have
the same dimension, and on a fixed finite-dimensional space any
two norms are equivalent (a theorem of Auerbach). So counterexamples
require infinite dimensions.
For c_0 itself the gap has been closed. Godefroy, Kalton and
Lancien (2000) proved:
* a Banach space bi-Lipschitz equivalent to c_0 is linearly
isomorphic to c_0;
* a Banach space bi-Lipschitz equivalent to a linear subspace
of c_0 is linearly isomorphic to a subspace of c_0.
Their theorems are about "the entire space of the equivalence".
They leave open a surgical question. Suppose Y contains a
bi-Lipschitz image of c_0 and Y is not c_0. Must Y contain a
linear copy of c_0? This was recorded by Kalton (2008, Problem 2)
and is answered negatively by the present work.
--------------------------------------------------------------------
5. THE MAIN THEOREM
Throughout, if X and Y are Banach spaces,
X (+) Y = X (+) Y with the sup norm
||(x,y)|| = max(||x||, ||y||).
("Direct sum.") Think of X (+) Y as "X next to Y, adding
coordinate by coordinate".
MAIN THEOREM. There exists a separable real Banach space Z such
that:
(i) Z contains no closed linear subspace linearly isomorphic
to c_0.
(ii) There is an onto bi-Lipschitz map
F: Z (+) c_0 -> Z.
"Separable" means Z has a countable dense subset, allowing
everything in Z to be approximated by countable data.
Consequences of the Main Theorem.
(a) Z (+) c_0 and Z are bi-Lipschitz equivalent (by F) but not
linearly isomorphic. (The direct sum contains a linear copy
of c_0; Z does not.)
(b) c_0 embeds bi-Lipschitzly into Z: restrict F to {0} (+) c_0.
Yet Z contains no linear copy of c_0. This gives a negative
answer to Kalton's Problem 2.
(c) Every separable metric space bi-Lipschitz-embeds into Z.
Aharoni's theorem (1974) already embeds such spaces into
c_0; composing with the restriction in (b) gives the claim.
So Z is a bi-Lipschitz-universal space that has no linear copy of
c_0 at all.
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6. WHY THIS IS SURPRISING
In finite dimensions, bi-Lipschitz geometry and linear structure
agree almost perfectly. Infinite-dimensional Banach spaces can be
bi-Lipschitz equivalent without being linearly isomorphic.
Aharoni and Lindenstrauss (1978) gave non-separable examples.
Ribe (1984) gave separable examples using uniform homeomorphism.
Aharoni and Lindenstrauss (1985) gave uniformly convex examples.
Johnson, Lindenstrauss and Schechtman (1996) produced many
further examples.
The 2000 Godefroy-Kalton-Lancien theorems killed a natural hope:
bi-Lipschitz equivalence is not all that far from linear
isomorphism when the entire space is c_0 or a subspace of c_0.
The open question Kalton isolated was surgical: could a
"crumpled" piece of c_0 hide inside a space that refuses any
linear copy of c_0? The answer we prove is yes.
--------------------------------------------------------------------
7. HOW THE PROOF GOES (INFORMAL)
The construction starts with an abstract algebraic observation.
Suppose E, U, Z are Banach spaces and we have three maps:
g: E (+) U -> U bi-Lipschitz and onto,
Q: Z -> U bounded linear,
K: E (+) U -> Z Lipschitz,
satisfying the exact identity
Q K = g - P, P(t, u) = u.
Then
F: Z (+) E -> Z, F(b, t) = b + K(t, Qb)
is a bi-Lipschitz bijection. Its inverse is
G(d) = ( d - K(t_d, u_d), t_d ),
where (t_d, u_d) = g^{-1}(Qd).
This is the absorption criterion: it reduces the whole theorem to
constructing three things.
Step 1 (build g). Take E = c_0(N) and U = c_0(union of blocks).
On each block, keep a "control scalar" from the previous block
and forget one of the block's own scalars. The choice of the
control direction is a state-dependent pair (v, w) with w v = 1.
Since w v = 1, the assignment is invertible when the state is
frozen, and the state-dependence is slow enough to preserve
bi-Lipschitz bounds globally. An extra terminal scalar at the
end of each finite row lets a Brouwer fixed-point argument
prove g is onto.
Step 2 (build Z). Z is the completion of mass-zero "molecules"
(formal finite sums of points in B = closed unit ball of D with
total mass zero). The norm on Z is a double sum (over "test
classes" and "operation budgets") of squares of suprema of
selected tests. It is engineered to have two properties:
(i) every individual test appears as a bounded linear
functional on Z, so the shift correction
h(s) = (S_s - P)s can be expected to be bounded-linear
in the molecule;
(ii) a hypothetical linear c_0 in Z would force the norms of
finite signed sums of "basis vectors" to grow like
sqrt(number of vectors), which is impossible in a c_0.
Step 3 (assembly). Q is the map that extracts the shift
correction from a molecule. K is a radial Lipschitz lift of the
map s -> [delta_s - delta_0]. The absorption criterion assembles
F from Q, K and g.
Section 1 gives the historical story in more detail. Section 2
proves the absorption criterion. Section 3 builds the weights.
Section 4 yields the onto shift g. Section 5 produces the test
functions. Section 6 builds Z and the maps Q and K. Section 7
rules out a linear c_0 in Z. Section 8 collects everything.
====================================================================
SECTION 1. INTRODUCTION
====================================================================
This section records the history and states the main theorem
precisely. All Banach spaces here are real.
1.1 SETUP AND HISTORY
A map F: X -> Y between Banach spaces is a bi-Lipschitz embedding
if there are constants
0 < c <= C < +inf, c ||x - x'||_X <= ||F(x) - F(x')||_Y
<= C ||x - x'||_X (x, x' in X).
F is a bi-Lipschitz equivalence if it is also surjective onto Y.
The two notions differ by surjectivity: an embedding lands in a
subset, an equivalence covers the whole target. This distinction
matters below: the final theorem constructs an onto map between
whole Banach spaces, while its linear-rigidity obstruction concerns
a subspace.
The nonlinear geometry of Banach spaces studies maps that control
distances but need not respect addition. This subject asks how much
linear structure survives a distortion-bounded change of coordinates.
Ribe (1976) proved that uniformly homeomorphic normed spaces have
the same finite-dimensional linear structure, up to one common
distortion bound [Ribe76]. Uniformly homeomorphic is weaker than
bi-Lipschitz equivalent: the modulus of continuity can be arbitrary
rather than linear.
Aharoni and Lindenstrauss (1978) constructed nonseparable Banach
spaces that are bi-Lipschitz equivalent but not linearly isomorphic,
and they asked for separable examples [AL78, pp. 281-282].
Ribe (1984) later produced separable uniformly homeomorphic
nonisomorphic spaces. Aharoni and Lindenstrauss (1985) obtained
uniformly convex examples that are uniformly homeomorphic but not
Lipschitz equivalent [Ribe84, AL85].
Johnson, Lindenstrauss and Schechtman (1996) developed both rigidity
results and separable examples with distinct linear structure under
the weaker relation of uniform homeomorphism [JLS96, Theorem 5.8
and Section 8(3)].
Collectively, this body of work explains why any separable
bi-Lipschitz counterexample to linear-isomorphism rigidity requires
more than a uniform deformation. The map has to thread a needle
between two related but incompatible structures.
1.2 THE ROLE OF c_0
The space c_0 = c_0(N) is the Banach space of real scalar
sequences tending to zero, under the supremum norm
||x||_{c_0} = sup_{n >= 1} |x_n|.
c_0 is the simplest infinite-dimensional sequence space that is
not Hilbert space. On the linear side it is completely classified:
its closed subspaces and quotients have explicit descriptions, and
its Banach-space properties (like not containing l_2 as a closed
subspace, and having the "extensible" property of being a
Lindenstrauss space) are well-understood.
Godefroy, Kalton and Lancien (2000) proved two rigidity results
about c_0 under bi-Lipschitz maps:
* A Banach space bi-Lipschitz equivalent to a linear subspace
of c_0 is linearly isomorphic to a subspace of c_0.
* A Banach space bi-Lipschitz equivalent to c_0 is linearly
isomorphic to c_0.
These are [GKL00, Theorems 2.1-2.2].
These two theorems concern the whole domain of a bi-Lipschitz
equivalence. They do not say that a Banach space containing a
bi-Lipschitz copy of c_0 must contain a linear copy of c_0. The
following surgical question was left open:
Let Y be a Banach space that contains a bi-Lipschitz image
of c_0. Must Y contain a closed linear subspace isomorphic
to c_0?
Kalton (2008) recorded this as Problem 2 [Kalton08, pp. 21-22],
separately from the separable Lipschitz-isomorphism question
(Problem 3). Hajek, Johanis and Schlumprecht, in their January 2024
preprint (arXiv:2401.00831v1) later published in 2025, still
described the embedding question as apparently open [HJS25].
The result proved here answers Kalton's Problem 2 negatively, and
by a stronger route: it does not just embed c_0 in some exotic
space, it arranges a bi-Lipschitz equivalence of that space with
its own direct sum with c_0.
1.3 DIRECT SUMS
For Banach spaces X, Y, the direct sum
X (+) Y
is the set of pairs (x, y) with x in X, y in Y, under coordinatewise
operations and the supremum norm
||(x, y)||_{X (+) Y} = max(||x||_X, ||y||_Y).
This is the c_0-sum: it adds two Banach spaces in the category of
c_0-type sums. A linear copy of c_0 sits inside X (+) c_0 as
{0} (+) c_0 = {(0, t) : t in c_0}.
1.4 THE MAIN THEOREM
THEOREM 1.1 (Main). There is a separable real Banach space Z
containing no closed linear subspace isomorphic to c_0, and
an onto bi-Lipschitz map
F: Z (+) c_0 -> Z.
We will prove this in Section 8, after building Z and F in
Sections 3-7.
Three immediate observations.
First, F is automatically a bijection. Its lower Lipschitz bound
forbids F(b,t) = F(b',t') unless (b,t) = (b',t').
Second, Z (+) c_0 and Z cannot be linearly isomorphic. Z (+) c_0
contains a linear copy of c_0 as {0} (+) c_0. A bounded linear
isomorphism from Z (+) c_0 onto Z would carry this subspace to a
closed linear copy of c_0 inside Z, contradicting (i).
Third, restricting F to the summand {0} (+) c_0 gives a bi-Lipschitz
embedding
c_0 -> Z, t |-> F(0, t).
This is a bi-Lipschitz image of c_0 inside Z, while Z contains no
linear copy of c_0. This makes Z a counterexample to Kalton's
Problem 2.
Aharoni (1974) proved that every separable metric space embeds
bi-Lipschitzly into c_0 [Aharoni74]. Composing such an embedding
with the map c_0 -> Z just described makes Z a bi-Lipschitz
universal target for all separable metric spaces.
1.5 RELATED CONSTRUCTIONS
Three earlier or concurrent constructions have weaker control.
Kalton (2004) constructed a separable Schur space S (weakly
convergent sequences converge in norm) that is uniformly
homeomorphic to a space containing a complemented copy of c_0
[Kalton04, Theorem 4.6 and Propositions 5.1-5.2]. The Schur
property excludes a linear copy of c_0. Interleaving the two c_0
summands gives uniform c_0 absorption by that same Schur space,
with a nonlinear modulus of continuity.
Sari (2026) constructed a coarse-Lipschitz embedding of c_0 into
a separable dual Banach space that contains no linear copy of c_0
[Sari26, Introduction and Theorem 5]. Coarse-Lipschitz distance
bounds allow additive error. Theorem 1.1 achieves global two-sided
Lipschitz bounds and an onto map, which is strictly stronger.
Independently of these, the OpenAI 2026 preprint [OpenAI2026Baseline]
constructed a real-separable counterexample to the Lipschitz-
isomorphism question using a different linear obstruction. In
that construction one space contains c_0(l_2), the space of
l_2-valued null sequences, while its bi-Lipschitz equivalent
partner contains no linear copy of c_0(l_2). The Hilbert-block
and localized Lipschitz-free argument there is independent of the
construction here; the present construction excludes ordinary c_0
and yields absorption by the same space.
1.6 PROOF OVERVIEW
The proof splits into an algebraic reduction (Section 2) and
three concrete constructions (Sections 3-7). The reduction is
elementary and independent of coordinates; it tells us exactly
what the concrete construction must supply.
ALGEBRAIC REDUCTION. Suppose E, U are Banach spaces, and P is the
projection from E (+) U onto U:
P: E (+) U -> U, P(t, u) = u.
We want an onto bi-Lipschitz map g: E (+) U -> U; a bounded linear
map Q: Z -> U; and a Lipschitz map K: E (+) U -> Z, subject to
the exact identity
Q K = g - P. (*)
Section 2 proves that any such (g, Q, K) yields an onto bi-Lipschitz
map
F: Z (+) E -> Z, F(b, t) = b + K(t, Qb).
Then taking E = c_0 and keeping Z with no linear copy of c_0
delivers Theorem 1.1.
The identity (*) is what makes this work. The identity map on E
supplies a lift of the correction g - P, without any surjectivity
or right-inverse condition on Q. This is the key point: a
Lipschitz right inverse of a linear quotient onto a separable
Banach space always yields a bounded linear right inverse
[GK03, Corollary 3.2], so insisting on surjectivity of Q would
destroy the possibility of a genuinely nonlinear construction.
Requiring only (*) keeps the map Q potentially non-surjective.
CONCRETE TASK 1: build g. Take E = c_0(N) and U a countable-
coordinate copy of c_0 built out of finite blocks. Section 3
constructs finite-coordinate weights v^λ(s), w^λ(s) and their
localized correction functions f_γ. Section 4 arranges the blocks
of U into finite rows. In each block choose a vector v and a
row-functional w with w v = 1. A scalar entering from the
preceding block replaces the scalar readout w x of the current
input block, while the rest of the block is unchanged. Denote this
state-s-dependent frozen shift S_s. With all v, w fixed, the
output readouts recover the input via an explicit linear inverse.
The weights actually depend on the normalized input, so that
frozen inverse does not by itself invert the nonlinear map.
CONTROLLING THE WEIGHT DEPENDENCE. Section 4 controls the change
of the weights to get global bi-Lipschitz bounds, then uses
Brouwer's fixed-point theorem on finite rows to prove onto-ness.
An extra terminal scalar at the end of each row balances
dimensions and forces the possible output spillover to vanish.
Density, plus a closed-image argument, upgrades onto-ness from
finite rows to the whole of U.
CONCRETE TASK 2: build Z and lift the correction. Record each
coordinate of (S_s - P)s as a scalar function of s in the unit
ball B = {s in E (+) U : ||s||_{E (+) U} <= 1}. These functions
depend on finitely many coordinates and have localized supports.
Section 5 enlarges them by two local operations:
* one makes a function constant on a small cylinder, and
* one retains only the change of a function over a small
cylinder.
A strict scale budget keeps their supports localized, and a
nonincreasing order of scales controls what happens after a small
local change. Tail factors give a second way to separate supports.
Section 6 pairs these tests with mass-zero finite combinations
of points ("molecules"), following Arens and Eells [AE56,
Section 2] and the Lipschitz-free framework of Godefroy and Kalton
[GK03, Sections 1-2]. The norm is built from lists of selected
tests whose signed sums have a common Lipschitz bound. It combines
their evaluation norms, with exponents approaching one, in an
outer square sum. The square sum prevents a hypothetical c_0
basis from escaping through different test classes; the varying
exponents then select one scalar test detecting each basis vector.
Section 6 proves directly the completion and duality properties
needed for this norm. The coordinate tests define Q, and radial
extension of the point molecules defines K so that (*) holds.
CONCRETE TASK 3: no linear c_0 in Z. A hypothetical linear c_0
basis in Z would give vectors with uniformly bounded signed sums.
Section 7 extracts one detecting test for each vector from a single
class with a common operation budget. A test's label records its
original correction coordinate. Distinct labels can be separated
either by tails or by nested coordinate cylinders. For a repeated
label, an expansion leaves either disjoint tails or a first small
local change. In the latter case, its support shrinks and its
value becomes small enough that nested tests still have uniformly
Lipschitz signed sums. The norm then forces the signed sums of
the vectors to grow, a contradiction.
ASSEMBLY. Section 8 combines the obstruction with (*) and the
onto shift to prove Theorem 1.1 and its three same-space
consequences.
====================================================================
SECTION 2. THE ABSORPTION CRITERION
====================================================================
The passage from a lifted correction to an absorption map is
elementary and independent of the coordinate construction. We
state it first in order to specify exactly what that construction
must supply.
Triangular changes of variables also appear in the quotient-
section construction of Aharoni and Lindenstrauss [AL78, pp.
281-282] and in the free-space splitting of Godefroy and Kalton
[GK03, Theorem 2.12]. The following argument lifts a correction
term and supplies its own inverse formula.
2.1 THE LEMMA
LEMMA 2.1 (Absorption from a lifted correction). Let E, U, Z
be real Banach spaces. Put
D = E (+) U,
with the sup norm ||(t,u)||_D = max(||t||_E, ||u||_U), and let
P: D -> U, P(t, u) = u,
be the projection. Suppose:
* Q: Z -> U is bounded linear,
* g: D -> U is a bi-Lipschitz bijection,
* K: D -> Z is Lipschitz,
and the exact identity
Q K(t, u) = g(t, u) - u for every (t, u) in D.
holds. Then
F: Z (+) E -> Z, F(b, t) = b + K(t, Qb)
is a bi-Lipschitz bijection.
Suppose additionally that
q = ||Q||,
L_K = Lip(K), and
||g(x) - g(x')||_U >= c_g ||x - x'||_D for some c_g > 0.
Then
Lip(F) <= 1 + L_K max(1, q),
Lip(F^{-1}) <= max( 1 + L_K q / c_g, q / c_g ).
No surjectivity and no right-inverse assumption is imposed on Q.
2.2 PROOF OF INJECTIVITY AND SURJECTIVITY
For d in Z, define
(t_d, u_d) = g^{-1}(Qd),
G(d) = ( d - K(t_d, u_d), t_d ).
Every d gives a well-defined G(d): the map g is onto U by
hypothesis, so t_d and u_d exist for every d.
The identity QK = g - P gives
Q F(b, t) = Qb + QK(t, Qb) = g(t, Qb).
So g^{-1} reads (t, Qb) off QF(b,t):
g^{-1}( Q F(b,t) ) = (t, Qb).
Apply G to F(b,t):
G(F(b,t)) = ( b + K(t, Qb) - K(t_d, u_d), t_d )
with (t_d, u_d) = g^{-1}(QF(b,t)) = (t, Qb). Thus the K terms
cancel and the output is (b, t):
G(F(b,t)) = (b, t).
For the opposite composition, fix d in Z, write (t, u) = g^{-1}(Qd)
and set b = d - K(t, u). Then
Qb = Qd - QK(t, u)
= g(t, u) - (g(t, u) - u)
= u.
Here we used QK(t,u) = g(t,u) - u for the pair (t,u). Therefore
F(G(d)) = F(b, t) = b + K(t, Qb) = (d - K(t, u)) + K(t, u) = d.
This proves F is a bijection with inverse G.
2.3 LIPSCHITZ BOUNDS
UPPER BOUND. The map (b, t) -> (t, Qb) from Z (+) E to D is
Lipschitz with constant at most max(1, q): the second coordinate
is bounded by q times ||b||_Z, and the first coordinate is ||t||_E.
Both are bounded by ||(b, t)||_{Z (+) E}. K is L_K-Lipschitz,
the identity b -> b is 1-Lipschitz, and the sum of the two pieces
gives
||F(b,t) - F(b',t')||_Z
<= ||b - b'||_Z + ||K(t, Qb) - K(t', Qb')||_Z
<= ||b - b'||_Z + L_K ||(t - t', Q(b - b'))||_D
<= (1 + L_K max(1, q)) ||(b,t) - (b',t')||_{Z (+) E}.
LOWER BOUND. It is cleaner to bound the inverse G. Fix d, d' in Z
and put (t_d, u_d) = g^{-1}(Qd), (t_d', u_d') = g^{-1}(Qd'). The
lower bound for g gives
||(t_d, u_d) - (t_d', u_d')||_D <= (q / c_g) ||d - d'||_Z.
The two coordinates of G(d) are d - K(t_d, u_d) and t_d. Their
Lipschitz dependence on d is:
||(d - K(t_d, u_d)) - (d' - K(t_d', u_d'))||_Z
<= ||d - d'||_Z + L_K ||(t_d, u_d) - (t_d', u_d')||_D
<= (1 + L_K q / c_g) ||d - d'||_Z,
||t_d - t_d'||_D
<= (q / c_g) ||d - d'||_Z.
The maximum of the two coordinate bounds is
max( 1 + L_K q / c_g, q / c_g ).
That is the asserted upper Lipschitz constant for F^{-1}, and its
reciprocal is a lower Lipschitz constant for F. The proof of
Lemma 2.1 is complete.
2.4 WHAT REMAINS TO BE BUILT
The remaining sections supply this criterion with E = c_0. The
inverse calculation above uses only QK = g - P. It never asks
for an element b with prescribed Qb until the formula
b = d - K(g^{-1}(Qd)) has already produced one. This is the
essential feature that lets the construction avoid needing Q to
be surjective or to have a Lipschitz right inverse, which
[GK03, Corollary 3.2] would force to be a bounded linear
right inverse of a linear quotient.
====================================================================
SECTION 3. COORDINATE WEIGHTS AND LOCALIZED CORRECTIONS
====================================================================
The shift will replace one scalar in each coordinate block by a
scalar from the preceding block. We first describe this
operation, then choose weights whose dependence on the input is
slow and whose correction coordinates have the support properties
needed for the later tests.
3.1 LEVELS, BLOCKS AND RADII
For a countably infinite set A, let c_0(A) have its supremum
norm. Write
N = {1, 2, 3, ...},
index levels by λ = (i, n) in N^2, and put
J_λ = i + n.
For j ≥ 1 set
d_j = 2^{-j}, A_j = ( [-1, 1] ∩ d_j Z )^j.
For j = 0, let A_0 consist of one point (the empty tuple).
The finite coordinate block at level λ is
Γ_λ = { (λ, j, ξ) : 0 ≤ j ≤ J_λ, ξ ∈ A_j }.
We work in the following three Banach spaces:
E = c_0(N),
U = c_0( disjoint union over λ in N^2 of Γ_λ ),
D = E (+) U.
Write x = (t, (x^λ)_λ) for an element of D, with t in E and
x^λ in R^{Γ_λ} for each λ. Put P(t, u) = u.
Fix an enumeration of the disjoint union of the coordinate sets
of E and U. The map
π_j: D -> R^j
retains the first j coordinates in this enumeration; π_0 is the
unique map into R^0.
Write
B = { s in D : ||s||_D ≤ 1 }
for the closed unit ball of D. For n > 1 let λ^- = (i, n-1) and
put
r_{i,1}(x) = max( |t_i|, ||x^{i,1}||_∞ ),
r_{i,n}(x) = max( ||x^{i,n-1}||_∞, ||x^{i,n}||_∞ ).
Each r_λ is 1-Lipschitz. Every input coordinate occurs in at most
two of these maxima. Therefore, for every fixed x in D,
r_λ(x) -> 0 as λ leaves finite sets. (3.1)
Indeed, infinitely many radii at least ε > 0 would require
infinitely many input coordinates of modulus at least ε, and
c_0-type spaces only have finitely many coordinates of modulus
≥ ε above any fixed threshold.
3.2 THE SCALAR REPLACEMENT
The shift will replace one scalar in each block by an incoming
scalar from the preceding block. To do that we need a "control
direction" inside each block. Fix a vector
v in R^{Γ_λ}
and a row (a linear functional)
w in (R^{Γ_λ})^*
with
w v = 1.
For a block vector z in R^{Γ_λ} and an incoming scalar b in R,
z = (z - v w z) + v (w z),
z + v(b - w z) = (z - v w z) + v b.
The first summand belongs to ker w, so the operation
z |-> z + v(b - w z)
preserves ker w and replaces the scalar readout w z by b. Along a
row of blocks we will take b from the preceding block's readout,
with t_i entering the first block.
The shift in Section 4 uses this algebraic identity when the
vectors and rows are frozen. In the actual construction, v and w
depend on the state s. The two conditions below combine the
identity w v = 1 with a bound on the variation of (v, w):
(W1) ||v^λ(s)||_∞ ≤ 1, ||w^λ(s)||_1 ≤ 2,
w^λ(s) v^λ(s) = 1.
(W2) For s, s' in B, put m = min(r_λ(s), r_λ(s')). Then
m * max( ||v^λ(s) - v^λ(s')||_∞,
||w^λ(s) - w^λ(s')||_1 )
≤ η ||s - s'||_D,
where η = 10^{-3}.
(W2) says the pairs vary slowly after multiplication by the
local radius, which keeps the size of the relevant blocks
bounded. This is the estimate that preserves the shift's
distance bounds when the weights vary with the input.
The bands j in each block serve a different purpose:
localization. For j ≥ 1, a correction coordinate indexed by
(λ, j, ξ) will be supported where the first j state coordinates
are close to ξ. Whenever j < J_λ, it will also require a positive
lower bound on r_λ(s) depending only on j. The first restriction
becomes sharper as j grows; the second gives a fixed radial
threshold when j is fixed and the level varies. We will prove
these two support bounds for the correction coordinates after
building the weights.
3.3 WEIGHTS AT ONE LEVEL
LEMMA 3.1 (Weights at one level). Let η = 10^{-3}. There are
continuous maps
v^λ: B -> R^{Γ_λ},
w^λ: B -> (R^{Γ_λ})^*,
depending on finitely many coordinates of their argument,
such that (W1) and (W2) hold. More precisely, the j-th band
of either weight is the product of a coefficient
θ_j^{J_λ}( r_λ(s) ) and a vector or row that depends only on
π_j s. If θ_j^{J_λ}(r) > 0 then
r ≤ ρ_j (j ≥ 1),
r ≥ ℓ_j (j < J_λ), (3.2)
for positive constants ρ_j, ℓ_j defined in the proof.
PROOF OF LEMMA 3.1 (construction of the bands).
Fix j ≥ 1 and s in B. For ξ in A_j put
b^j_ξ(s) = ( 1 - ||π_j s - ξ||_∞ / d_j )_+,
B_j(s) = sum_{ξ in A_j} b^j_ξ(s).
The grid A_j includes both endpoints in every coordinate (and
hence every boundary point of the cube). Every point of [-1, 1]^j
is within d_j / 2 in supremum norm of a grid point. Thus
B_j(s) ≥ 1/2 for every s in B,
and the following rows are defined everywhere on B:
w^j_ξ(s) = b^j_ξ(s) / B_j(s),
v^j_ξ(s) = min( 1, ( 2 - ||π_j s - ξ||_∞ / d_j )_+ ).
The entries are nonnegative. Moreover
||w^j(s)||_1 = 1, ||v^j(s)||_∞ ≤ 1.
If w^j_ξ(s) > 0, then ||π_j s - ξ||_∞ < d_j and hence
v^j_ξ(s) = 1. Consequently,
w^j(s) v^j(s) = 1.
At band j = 0, set both the sole coordinate of v^0 and the sole
entry of w^0 equal to 1.
EXPLICIT LIPSCHITZ BOUNDS. Put
N_j = |A_j| = (2^{j+1} + 1)^j.
Each b^j_ξ and v^j_ξ has Lipschitz constant at most d_j^{-1}, and
there are N_j of them, so
||b^j(s) - b^j(s')||_1 ≤ N_j d_j^{-1} ||s - s'||_D.
Normalizing a nonnegative vector whose l_1 norm is at least 1/2
changes pairwise distances by at most a factor 4 in l_1:
|| b/||b||_1 - b'/||b'||_1 ||_1
≤ ( ||b - b'||_1 + | ||b||_1 - ||b'||_1 | ) / ||b||_1
≤ 4 ||b - b'||_1.
Thus
L_j = 4 N_j / d_j
bounds both Lip(w^j) in l_1 and Lip(v^j) in the supremum norm.
CONTINUATION OF THE PROOF OF LEMMA 3.1 (theta schedule).
Choose H > 0 with π/H ≤ η/2, and inductively choose numbers
ρ_j > 0 so that
0 < ρ_1 < 1,
ρ_{j+1} < e^{-H} ρ_j,
L_j ρ_j ≤ η/4 (j ≥ 1).
Each constraint at stage j has a positive right-hand side, so
this choice is possible. Put
ℓ_j = e^{-H} ρ_{j+1} (j ≥ 0).
For an integer J ≥ 1 define
θ^J(r) = ( θ^J_0(r), θ^J_1(r), ..., θ^J_J(r) )
on [0, 1] as follows.
* For r ≥ ρ_1 set θ^J(r) = e_0 (the first standard basis
vector of R^{J+1}).
* As r decreases across the interval [e^{-H} ρ_j, ρ_j], for
1 ≤ j ≤ J, rotate from e_{j-1} to e_j by setting
θ^J_{j-1}(r) = cos( α_j(r) ),
θ^J_j(r) = sin( α_j(r) ),
α_j(r) = (π / (2H)) * log( ρ_j / r ),
with all other entries zero.
* Between these transition intervals, keep the most recently
reached unit vector.
* For 0 ≤ r ≤ e^{-H} ρ_J set θ^J(r) = e_J.
The strict inequality ρ_{j+1} < e^{-H} ρ_j makes the transition
intervals disjoint and ordered. At each transition endpoint the
formula agrees with the adjacent constant value; this also gives
continuity at r = 0. Every entry is nonnegative, and
sum_{j=0}^{J} θ^J_j(r)^2 = 1, (3.3a)
sum_{j=0}^{J} θ^J_j(r) ≤ 2. (3.3b)
For 0 < r ≤ 1 the path u -> θ^J(e^u) is piecewise continuously
differentiable and has l_1 speed at most
(π / (2H)) * ( |sin α| + |cos α| ) ≤ π/H.
It is therefore π/H-Lipschitz in the variable log r. The
schedule above also yields (3.2). For j = 0, only the second
line of (3.2) applies (no upper restriction on r). For j = J,
only the first line applies, and θ^J_J(0) = 1 is allowed.
For J = J_λ and r = r_λ(s), now define the concatenated weights
v^λ(s) = ( θ^J_j(r) v^j(s) )_{j=0}^{J},
w^λ(s) = ( θ^J_j(r) w^j(s) )_{j=0}^{J}.
Every band depends on finitely many coordinates: π_j s and
r_λ(s) involve finite coordinate sets. Continuity includes
r = 0 and transition endpoints. The bounds on the band weights
and (3.3) give
||v^λ(s)||_∞ ≤ 1,
||w^λ(s)||_1 = sum_{j=0}^{J} θ^J_j(r) ≤ 2,
w^λ(s) v^λ(s) = sum_{j=0}^{J} θ^J_j(r)^2 = 1.
This proves (W1).
For (W2), put
r = r_λ(s), r' = r_λ(s'),
m = min(r, r'), δ = ||s - s'||_D.
If m = 0 there is nothing to prove. Otherwise
m |log r - log r'| ≤ |r - r'| ≤ δ.
Change the band coefficients first, using the band weights at
s'. The logarithmic Lipschitz estimate and the bounds
||v^j(s')||_∞ ≤ 1 and ||w^j(s')||_1 ≤ 1 show that this part of
either norm difference, multiplied by m, is at most
(π/H) δ ≤ (η/2) δ.
Then change the band weights with coefficients frozen at
θ^J(r). Only positive j vary, and θ^J_j(r) > 0 implies
m ≤ r ≤ ρ_j. Hence this part of either norm difference,
multiplied by m, is at most
δ * sum_{j=1}^{J} θ^J_j(r) * m L_j
≤ (η δ / 4) * sum_{j=1}^{J} θ^J_j(r)
≤ (η δ / 2).
The sum of the two contributions proves (W2). This completes
the proof of Lemma 3.1.∎
3.4 CORRECTION COORDINATES
For s in B (the closed unit ball of D) define the readout
scalars
a_{i,0}(s) = t_i,
a_{i,n}(s) = w^{i,n}(s) x^{i,n},
where λ = (i, n) is a level, t_i is the i-th coordinate of the
E-part of s, and w^{i,n}(s) is the weight row at level (i,n)
evaluated on the U-part x^{i,n} of s.
For γ = (λ, j, ξ), where λ = (i, n) and 0 ≤ j ≤ J_λ and
ξ ∈ A_j, define the "correction coordinate"
f_γ(s) = θ_j^{J_λ}( r_λ(s) ) * v^j_ξ(s)
* ( a_{i,n-1}(s) - a_{i,n}(s) ). (3.4)
These f_γ will be exactly the coordinate functions of the
frozen shift correction (S_s - P)s from Section 4. Preview: the
frozen shift is (S_s x)^λ = x^λ + v^λ(a_{λ^-} - a_λ), so
(S_s - P) x has λ-th coordinate v^λ(a_{λ^-} - a_λ). Distributing
v^λ into its bands via v^λ = ( θ_j^J v^j )_{j=0}^J, and
evaluating at x = s, gives exactly f_γ(s) in the γ-coordinate
of U. So each f_γ is one coordinate of the correction.
The next lemma collects everything we will need about f_γ.
LEMMA 3.2 (Finite dependence and local support). For every
γ = (λ, j, ξ), the function f_γ is continuous, vanishes at 0,
and depends on a nonempty finite set I_γ of coordinates of D.
The sets I_γ can be chosen to include the first j coordinates
of D and every coordinate used by r_λ. Uniformly in γ,
|f_γ(s)| ≤ 4 r_λ(s) ≤ 4, (3.5a)
Lip(f_γ) ≤ 4 + 12η. (3.5b)
Moreover,
f_γ(s) ≠ 0 implies ( ||π_j s - ξ||_∞ ≤ 2 d_j
if j ≥ 1,
r_λ(s) ≥ ℓ_j
if j < J_λ ), (3.6)
where both conditions are imposed when both apply.
PROOF of Lemma 3.2.
FINITE DEPENDENCE.
For n > 1, a_{i,n-1} uses only
s^{i,n-1}, r_{i,n-1}(s), π_{J_{i,n-1}} s.
The former radius uses s^{i,n-2} when n > 2, and t_i when n = 2.
The current scalar a_{i,n} uses only
s^{i,n}, r_{i,n}(s), π_{J_{i,n}} s.
For n = 1 the predecessor scalar is t_i.
Since J_{i,n-1} < J_{i,n}, one possible common finite set I_γ for
all coordinates γ at level λ = (i,n) is the union of:
* the first J_λ enumerated coordinates of D,
* the coordinate t_i,
* every coordinate of the blocks Γ_{i,k} with
k in { n, n-1, n-2 } ∩ N.
Call this union I_γ. It is nonempty and contains every coordinate
required by the statement.
BOUND |f_γ| ≤ 4 r_λ(s).
The row bound (W1) gives
| a_{i,n-1}(s) - a_{i,n}(s) | ≤ 4 r_λ(s),
also when n = 1. Hence
|f_γ(s)| ≤ 4 r_λ(s) ≤ 4, and f_γ(0) = 0.
UNIFORM LIPSCHITZ BOUND.
Fix s, s' in B and write δ = ||s - s'||_D. At each level λ choose
the frozen pair (v^λ, w^λ) from whichever of s, s' has the larger
local radius r_λ, breaking ties arbitrarily. If e ∈ {s, s'} is
the other endpoint, (W2) gives
r_λ(e) * max{ ||v^λ(e) - v^λ||_∞, ||w^λ(e) - w^λ||_1 } ≤ η δ.
The same bound is trivially true if e supplied the frozen pair.
For e = s, s', replacing its weights by the frozen weights inside
the correction vector v^λ(e)( a_{i,n-1}(e) - a_{i,n}(e) ) costs at
most 4ηδ from the vector and at most 2ηδ from its current and
predecessor rows taken together (at n = 1 the predecessor t_i is
unchanged).
With all weights frozen, the correction vector is linear in the
input and has norm at most
4 r_λ(s - s') ≤ 4 δ.
Comparing the two endpoints therefore gives
|f_γ(s) - f_γ(s')| ≤ (4 + 12η) δ.
SUPPORT.
Suppose f_γ(s) ≠ 0. In the product (3.4) both factors
θ_j^{J_λ}(r_λ(s)) and v^j_ξ(s) are positive. For j ≥ 1, the
formula for v^j_ξ gives
||π_j s - ξ||_∞ < 2 d_j,
which implies the first closed inequality in (3.6). If j < J_λ,
the second inequality follows from (3.2). This covers j = 0 as
well: there the grid condition is dropped, but the lower bound
r_λ(s) ≥ ℓ_0 still holds. This completes the proof of Lemma
3.2.∎
REMARKS ON THE CONSTANTS. From now on set
A := sup_γ ||f_γ||_∞ ≤ 4,
L_0 := sup_γ Lip(f_γ) ≤ 4 + 12η = 4.012. (3.7)
These bounds are uniform in γ. Because η = 10^{-3}, the
constant 4 + 12η is comfortably less than 4 + 1/80.
The lemma also pins down exactly where each f_γ can be nonzero:
on a small coordinate cylinder in the first j coordinates (when
j ≥ 1), intersected with an annular region in the local radius
(when j < J_λ). This locality is what the constructions in
Sections 4-7 will exploit.
====================================================================
SECTION 4. THE STATE-DEPENDENT SHIFT IS ONTO
====================================================================
The weights of Lemma 3.1 now provide the map required by the
absorption criterion. We first invert arbitrary frozen weights,
then bound their variation, and finally solve every target by a
finite-dimensional fixed-point argument.
4.1 FROZEN SHIFTS
First ignore the dependence on s. The identity w v = 1 supplies
an inverse for any independently frozen choice of the pairs.
LEMMA 4.1 (Frozen shift). Choose for every (i, n) a pair
v^{i,n}, w^{i,n} satisfying (W1), independently between
blocks. Define S: D -> U by
a_{i,0} = t_i,
a_{i,n} = w^{i,n} x^{i,n},
(Sx)^{i,n} = x^{i,n} + v^{i,n}( a_{i,n-1} - a_{i,n} ). (4.1)
Then S is a bounded linear bijection with
||S|| ≤ 5, ||S^{-1}|| ≤ 5.
PROOF of Lemma 4.1.
BOUND ||S|| ≤ 5.
The row bound (W1), |w^{i,n} y| ≤ ||w^{i,n}||_1 ||y||_∞ ≤ 2||y||_∞,
applies at every level. Hence
| a_{i,n-1} - a_{i,n} | ≤ 4 r_{i,n}(x),
also when n = 1 (the predecessor t_i satisfies |t_i| ≤ r_{i,1}(x)).
Then
||(Sx)^{i,n}||_∞ ≤ ||x^{i,n}||_∞ + 4 r_{i,n}(x)
≤ 5 r_{i,n}(x).
By (3.1), the block norms r_{i,n}(x) tend to zero as λ goes to
infinity, so Sx is in U and ||S|| ≤ 5.
FROZEN INVERSE.
For y in U, put
b_{i,n} = w^{i,n} y^{i,n},
and define
t_i = b_{i,1},
x^{i,n} = y^{i,n} - v^{i,n}( b_{i,n} - b_{i,n+1} ). (4.2)
The block norms of y tend to zero off finite sets: otherwise
infinitely many coordinates of y would stay away from zero.
Because |b_{i,n}| ≤ 2||y^{i,n}||_∞, the scalar family (b_{i,n})
belongs to c_0(N^2). Formula (4.2) then gives t_i ∈ c_0 and
block norms of x tending to zero. Because the blocks are finite,
this is x in D. The same estimates give
||x||_D ≤ 5 ||y||_U.
Applying w^{i,n} to (4.2) yields
w^{i,n} x^{i,n} = b_{i,n} - w^{i,n} v^{i,n} (b_{i,n} - b_{i,n+1})
= b_{i,n} - ( b_{i,n} - b_{i,n+1} )
= b_{i,n+1} = a_{i,n}.
So (4.1) gives S x = y. Conversely, when y = S x, applying
w^{i,n} to (4.1) gives
b_{i,n} = w^{i,n} y^{i,n} = w^{i,n} x^{i,n} = a_{i,n}.
But then a_{i,n} = b_{i,n} and the predecessor scalar
a_{i,n-1} = b_{i,n-1} for the (i, n) block matches b_{i,n-1} in
formula (4.2). Explicitly: (4.2) with y = Sx gives
x^{i,n} = (Sx)^{i,n} - v^{i,n}(b_{i,n} - b_{i,n+1})
= x^{i,n} + v^{i,n}(a_{i,n-1} - a_{i,n})
- v^{i,n}(b_{i,n} - b_{i,n+1})
= x^{i,n} + v^{i,n}(b_{i,n-1} - b_{i,n+1})
- v^{i,n}(b_{i,n} - b_{i,n+1})
= x^{i,n} + v^{i,n}(b_{i,n-1} - b_{i,n})
= x^{i,n}.
The formulas are therefore mutual inverses, and the estimate
||S^{-1}|| ≤ 5 above (which bounded ||x||_D by 5||y||_U) is the
inverse bound.∎
4.2 THE NONLINEAR MAP g
For s in B, let S_s be the frozen shift of Lemma 4.1 using
v^λ(s), w^λ(s) in every block. Define
g(0) = 0,
g(x) = S_{s_x}(x), s_x = x / ||x||_D, x ≠ 0. (4.3)
This map is positively homogeneous. Its weights are recomputed
from the input, so Lemma 4.1 alone does not invert g.
Rewriting (4.3) block by block:
( g(x) )^{i,n} = x^{i,n} + v^{i,n}(s_x)( a_{i,n-1} - a_{i,n} ),
where a_{i,0} = t_i and a_{i,n} = w^{i,n}(s_x) x^{i,n}. All
references to weights are evaluated at the state s_x, not at x
itself.
Along a "row" of blocks (fixed i, varying n), the scalar a_{i,n}
of block (i, n) is read by w^{i,n}(s_x). At the next block
(i, n+1), the incoming scalar b in the scalar replacement is
precisely this readout a_{i,n}. So the scalar readouts are
"shifted one block up" along the row. The first block (i, 1)
gets its incoming scalar from the E-part t_i.
4.3 THE SHIFT CORRECTION IS A RADIAL SCALING
The shift correction, i.e., the difference between g and the
projection P, is the part of g that records the "interference"
between neighboring blocks. It turns out to be a radial scaling
of the correction coordinates f_γ from Lemma 3.2.
For s in B, define
h(s) = ( S_s - P ) s. (4.4)
Lemma 4.1 guarantees h(s) ∈ U. By (3.4) the γ-coordinate of h(s)
is
( h(s) )_γ = f_γ(s).
Shift correction identity. For every x in D,
( g - P )(x) = ||x||_D * h( x / ||x||_D ) (x ≠ 0),
where P(x) is the U-part of x. In coordinates, for every
γ = (λ, j, ξ) and every x ≠ 0:
( ( g - P )(x) )_γ
= f_γ( x / ||x||_D ) * ||x||_D. (4.5)
Proof. Write s_x = x/||x||_D. Since S_{s_x} is
linear (Lemma 4.1),
g(x) = S_{s_x}(x) = ||x||_D * S_{s_x}( s_x ).
Also,
P(x) = ||x||_D * P( s_x ) = ||x||_D * s_x.
Therefore
( g - P )(x)
= ||x||_D * ( S_{s_x}(s_x) - P(s_x) )
= ||x||_D * ( S_{s_x} - P )( s_x )
= ||x||_D * h( s_x ).
Taking the γ-coordinate and invoking (4.4) gives (4.5).∎
Caveat: in (4.4)-(4.5), h(s_x) = (S_{s_x} - P) s_x is evaluated
at the state s_x (which has norm exactly one), not at x itself.
For arbitrary s in B with ||s|| < 1, h(s) = (S_s - P) s is still
well-defined, but P(s) does not need to be a fixed point of
S_s because s is not normalised.
4.4 GLOBAL DISTANCE BOUNDS
PROPOSITION 4.2 (Global distance bounds). For every x, x' in
D,
( 1/5 - 24η ) ||x - x'||_D
≤ ||g(x) - g(x')||_U
≤ ( 5 + 24η ) ||x - x'||_D. (4.6)
In particular, g is injective and continuous.
PROOF.
NORMALIZATION ESTIMATE. Suppose first that x, x' are both nonzero.
Put d = ||x - x'||_D and M = max( ||x||_D, ||x'||_D ). If
||x||_D = M, split x/M - x'/||x'||_D at x'/M and use the reverse
triangle inequality | ||x||_D - ||x'||_D | ≤ d:
|| x/M - x'/||x'||_D ||_D ≤ d/M + ||x'||_D * d / (M ||x'||_D)
= d/M + d/M
= 2d/M.
The other order (when ||x'||_D = M) is the same. In terms of
s_x and s_{x'}:
|| s_x - s_{x'} ||_D ≤ 2d / M. (4.7)
FREEZE AT EACH LEVEL. At each level λ, freeze the entire pair
(v^λ, w^λ) at whichever of s_x, s_{x'} has the larger r_λ,
breaking a tie arbitrarily. These choices may vary with λ, but
Lemma 4.1 still applies to the resulting frozen shift S̄:
S̄ is the frozen shift with pairs (v̄^λ, w̄^λ) at level λ. For
e in { s_x, s_{x'} }, the chosen pair v̄^λ either agrees with the
weight at s_e, or r_λ(s_e) is the smaller of the two radii.
Thus (W2) and (4.7) give
r_λ(s_e) max{ ||v^λ(s_e) - v̄^λ||_∞, ||w^λ(s_e) - w̄^λ||_1 }
≤ 2 η d. (4.8)
This remains valid when r_λ(s_e) = 0 (both sides are then zero
because v̄^λ is frozen exactly at s_e and the error is trivially
zero).
BLOCK-BY-BLOCK COMPARISON. Fix e in { x, x' } and consider the
block λ = (i, n). We compare the block λ of g(e) = S_{s_e}(e)
with the block λ of S̄(e). First change its vector v^λ (from
v^λ(s_e) to the frozen v̄^λ) while keeping the two scalars
computed with the original rows a_{i,n-1}(e) and a_{i,n}(e). Their
difference has modulus at most 4 r_λ(e), so (4.8) bounds this
change by
8 η d.
Changing the current row w^λ changes only its scalar readout
a_{i,n}(e), by at most 2ηd, because ||e^λ||_∞ ≤ r_λ(e). If n > 1,
changing the predecessor row w^{i,n-1} costs another 2ηd, using
(4.8) at (i, n-1) and ||e^{i,n-1}||_∞ ≤ r_{i,n-1}(e). For n = 1,
the predecessor t_i does not change. Since ||v̄^λ||_∞ ≤ 1, we have
|| ( S_{s_e} e - S̄ e )^λ ||_∞ ≤ 12 η d. (4.9)
Taking the supremum over λ and adding the two endpoint errors,
|| g(x) - g(x') - S̄(x - x') ||_U ≤ 24 η d. (4.10)
The two frozen bounds of Lemma 4.1 give
( 1/5 ) d ≤ || S̄(x - x') ||_U ≤ 5 d. (4.11)
Combining (4.10) and (4.11),
||g(x) - g(x')||_U ≤ 5d + 24ηd = (5 + 24η)d,
||g(x) - g(x')||_U ≥ (1/5)d - 24ηd.
The second inequality is (1/5 - 24η) d, because the frozen
lower bound gives ||S̄(x-x')||_U ≥ d/5, and g(x)-g(x') differs
from S̄(x-x') by at most 24ηd. This is (4.6).
EDGE CASES. If one endpoint is zero (say x' = 0), then by
Lemma 4.1 applied to S_{s_x},
||S_{s_x} x||_U ≥ (1/5) ||x||_D = (1/5) d,
||S_{s_x} x||_U ≤ 5 ||x||_D = 5d.
These are stronger than the constants in (4.6) with η, because no
endpoint comparison is needed. Finally, 1/5 - 24η > 0 because
η = 10^{-3}, so the lower bound is strictly positive. This
proves (4.6) in all cases.
INJECTIVITY AND CONTINUITY. If g(x) = g(x'), the lower bound
forces ||x - x'||_D = 0, so x = x'. The upper bound shows g is
Lipschitz, hence continuous.∎
4.5 ONTO-NESS VIA A BROUWER ARGUMENT
The lower bound (4.6) gives injectivity and closed image, but not
surjectivity. The obstruction is a possible finite target y for
which no preimage exists. This is where the extra terminal scalar
at the end of each finite row enters.
The idea. The map g "pushes the scalar readout of block (i,n)
into block (i,n+1)". For a finite target y in U with finitely many
nonzero blocks, we can read y's block scalars off and feed them
back, block by block, to solve for the input. The extra scalar at
the end of each row (i, n), n = 1, ..., N, is a "spare coordinate"
that is not consumed by the first N-1 blocks. Forcing it to zero
kills the only way the N-th block of g could "leak" outside the
N-square. Brouwer's fixed-point theorem supplies the existence of
such an x.
Finite truncations. For N >= 1, let U_N be the coordinate
subspace of U consisting of the blocks with 1 <= i, n <= N, all
other coordinates zero. Let D_N consist of these blocks together
with t_1, ..., t_N in E, all other input coordinates zero. Both
are finite-dimensional, and
dim D_N = N + sum_{i,n<=N} |Γ_{i,n}| = dim( U_N (+) R^N ). (4.12)
The two sides have the same dimension because the N extras R^N
correspond to the terminal scalars a_{i,N} for i <= N.
For s in B, define
S_s^{(N)}: D_N -> U_N (+) R^N,
S_s^{(N)} x = ( ((S_s x)^{i, n})_{i, n <= N}, (4.13)
( w^{i,N}(s) x^{i,N} )_{i <= N} ).
The last N coordinates retain the terminal scalars a_{i,N}.
Finite inverse. Given (y, (c_i)_{i <= N}) in U_N (+) R^N, define
b_{i,n} = w^{i,n}(s) y^{i,n},
a_{i,0} = b_{i,1},
a_{i,n} = b_{i,n+1} (n < N),
a_{i,N} = c_i,
and then
t_i = a_{i,0},
x^{i,n} = y^{i,n} - v^{i,n}(s)( a_{i,n-1} - a_{i,n} ) (n <= N). (4.14)
Because a_{i,n-1} = b_{i,n}, applying w^{i,n}(s) to (4.14) gives
w^{i,n}(s) x^{i,n}
= b_{i,n} - w^{i,n}(s) v^{i,n}(s) ( b_{i,n} - a_{i,n} )
= b_{i,n} - ( b_{i,n} - a_{i,n} )
= a_{i,n}.
Substituting back in (4.13) returns (y, c). Conversely, applying
the rows to (4.13) recovers the b_{i,n}, and (4.14) recovers every
input coordinate. So (4.14) is the inverse of (4.13).
Bounded both ways. The same estimates as for the frozen shift
give
||S_s^{(N)}|| <= 5, ||(S_s^{(N)})^{-1}|| <= 5. (4.15)
Explicitly: if h_0 = ||(y,c)||, then |b_{i,n}| <= 2h_0, all
a_{i,n} have modulus at most 2h_0, and (4.14) has block norm at
most h_0 + 4h_0 = 5h_0. So ||x||_D <= 5h_0. For the forward
bound, the block estimate is 5||x||_D and the terminal coordinates
are at most 2||x||_D.
Continuity in s. The finitely many weights appearing in (4.14)
are continuous in s. Hence
s |-> ( S_s^{(N)} )^{-1}
is continuous as a map from B into the finite-dimensional linear
operators between these finite-dimensional spaces.
Brouwer argument. Let y in U_N with y != 0, and set h = ||y||_U > 0.
On the closed ball
Ball_5h = { x in D_N : ||x||_D <= 5h },
define
s(x) = x / max( ||x||_D, h/6 ), (4.16)
T(x) = ( S_{s(x)}^{(N)} )^{-1} ( y, 0 ). (4.17)
The denominator in (4.16) is positive even at x = 0, so s(x) in B
depends continuously on x. Equations (4.14)-(4.15) show T is a
continuous self-map of this finite-dimensional closed ball.
Brouwer's fixed-point theorem gives a fixed point x = T(x).
At a fixed point,
h = ||(y, 0)|| = || S_{s(x)}^{(N)} x || <= 5 ||x||_D,
so ||x||_D >= h/5 > h/6. Therefore
s(x) = x / max( ||x||_D, h/6 ) = x / ||x||_D = s_x,
the true normalized state. On the N-square, (4.13) gives
(g(x))^{i,n} = y^{i,n} for i, n <= N, and the terminal
coordinates give a_{i,N} = 0 for every i <= N.
Outside the N-square. The input is zero outside D_N. The only
outside block with a possibly nonzero predecessor scalar is
(i, N+1) for i <= N. There
(g(x))^{i,N+1} = v^{i,N+1}(s_x) a_{i,N} = 0.
All other outside blocks have zero input and zero predecessor
scalar. Hence g(x) = y in U. The zero target is g(0), so
U_N subset g(D) for every N. (4.18)
Density and closed image. Every finitely supported element of U
belongs to some U_N, and such elements are dense in U. The image
g(D) is closed: if g(x_k) converges in U, the lower bound (4.6)
makes (x_k) Cauchy in D; its limit x satisfies g(x_k) -> g(x) by
continuity. So g(D) is both dense and closed in U. Therefore
g(D) = U. (4.19)
Combining with Proposition 4.2:
PROPOSITION 4.3 (Onto-ness of the nonlinear shift). The map
g: D -> U is onto. Together with Proposition 4.2 it is a
bi-Lipschitz bijection.
A small diagram of one finite row of length N. The "terminal
scalar" a_{i,N} is an extra target that no block consumes, except
possibly the (i, N+1) block outside the finite equation:
[a_{i,0}=t_i] -> [y^{i,1}] -> ... -> [y^{i,N}] -> [c_i = a_{i,N}]
w^{i,1}y^{i,1} w^{i,N}y^{i,N} extra
= a_{i,0} = a_{i,N-1} target
|
| x^{i,N+1} = 0
v
[(i, N+1) output]
[v^{i,N+1} c_i]
Brouwer is applied with c = 0, which forces x^{i,N+1} = 0 and
hence no output in the (i, N+1) block.
4.6 THE CORRECTION h(s) AND ITS CONSTANTS
Recall P: D -> U, P(t, u) = u. For s in B, put
h(s) = ( S_s - P ) s. (4.20)
The frozen lemma ensures h(s) in U; by (3.4) its γ-coordinate is
exactly f_γ(s).
A small warning. The map S_s s uses s itself as the state, even
when ||s||_D < 1, whereas g(s) uses the normalized state
s / ||s||_D. The lifting construction of Section 6 will use h on
B and g only on normalized states. In particular, h(s) need not
equal (g - P)(s); the two differ by a radial factor (see Lemma
4.2).
Bounds from Lemma 3.2. By (3.5) the correction coordinates have
common bounds
A := sup_γ || f_γ ||_∞ <= 4, (4.21)
L_0 := sup_γ Lip( f_γ ) <= 4 + 12η.
These constants will be used repeatedly in Section 5.
What is left. We now have an onto bi-Lipschitz g: D -> U
(Propositions 4.2-4.3), and the correction h(s) = (S_s - P)s that
g measures. The absorption criterion needs Q: Z -> U with
Qζ(s) = h(s) and a Lipschitz lift K: D -> Z with QK = g - P.
Sections 5-7 build the target Banach space Z, the bounded linear
map Q, and the Lipschitz lift K.
====================================================================
SECTION 5. SELECTED TESTS AND THEIR SUPPORTS
====================================================================
The goal is to construct a Banach space Z that captures the shift
correction h(s) = (S_s - P)s and rules out linear c_0. The
coordinate functions f_γ of (3.4) are too simple to build Z from;
each depends only on the coordinates of one block and its
predecessor. Section 5 enlarges this family of "tests" by two
local operations and a tail factor, while retaining quantitative
control over their supports and Lipschitz constants.
5.1 TWO LOCAL OPERATIONS
Fix a label γ = (λ, j, ξ) and its nonempty finite set I = I_γ of
coordinates of D. A function on B that depends only on I is
identified with its function on the cube [-1, 1]^I, evaluating at
the zero extension of a cube point. Write s|_I for coordinate
restriction.
For a dyadic scale ε = 2^{-a}, a in N_0 = {0, 1, 2, ...}, let
Q_{I,ε} = ( [-1, 1] ∩ (ε/8) Z )^I.
This is a finite grid that includes both endpoints in each
coordinate. Every point of the cube is within ε/16, hence within
ε/8, of a grid point in the supremum norm.
Define the 2-Lipschitz function
χ(t) =
0 0 ≤ t ≤ 1/2,
2t - 1 1/2 < t < 1,
1 t ≥ 1.
For a Lipschitz function u that depends on I, and q in Q_{I,ε}, put
(G_{q,ε} u)(s) = u(q) + χ( ||s|_I - q||_∞ / ε )( u(s) - u(q) ),
R_{q,ε} u = G_{q,ε} u - u. (5.1)
Both operations are linear in u, retain dependence on I, and
G_{q,ε} u is the constant u(q) on the open cylinder
||s|_I - q||_∞ < ε/2.
The two operations. G makes u locally constant on a small
cylinder; R records the change. The identity
G = Id + R
will separate the "fixed part" of a test from the "change at
small scales". Both behave well under Lipschitz estimates. Also,
evaluating at q: G_{q,ε} u(q) = u(q), because χ(0) = 0, so
R_{q,ε} u(q) = 0.
LEMMA 5.1 (Bounds and geometric support). For every such u,
q, ε:
||R_{q,ε} u||_∞ ≤ ε Lip(u), (5.2a)
Lip(R_{q,ε} u) ≤ 3 Lip(u), (5.2b)
Lip(G_{q,ε} u) ≤ 4 Lip(u), (5.2c)
||G_{q,ε} u||_∞ ≤ ||u||_∞, (5.2d)
||R_{q,ε} u||_∞ ≤ 2 ||u||_∞. (5.2e)
Let
K_I(u) = closure in [-1, 1]^I of { s|_I : s in B, u(s) != 0 }.
In the next display write G = G_{q,ε} and R = R_{q,ε}. With
K_ε = { p : dist_∞(p, K_I(u)) ≤ ε } and the convention that
∅_ε = ∅:
K_I(G u) , K_I(R u) ⊂ K_I(u)_ε, (5.2f)
K_I(R u) ⊂ closure B_∞( q, ε ). (5.2g)
PROOF of Lemma 5.1.
Set
a(s) = 1 - χ( ||s|_I - q||_∞ / ε ).
Then
R_{q,ε} u(s) = a(s)( u(q) - u(s) ),
0 ≤ a ≤ 1,
Lip(a) ≤ 2/ε.
SUPREMUM BOUNDS.
If a(s) != 0, then ||s|_I - q||_∞ < ε. The cylindrical Lipschitz
bound gives |u(q) - u(s)| ≤ ε Lip(u). This proves (5.2a), which
also implies (5.2e). And G = u + R is a convex combination of u(q)
and u(s):
(G_{q,ε} u)(s) = (1 - a(s)) u(s) + a(s) u(q),
so |G(s)| ≤ max( |u(s)|, |u(q)| ) ≤ ||u||_∞. This is (5.2d).
LIPSCHITZ BOUND.
Compare Ru(s) with Ru(t). If both a(s) and a(t) are zero there is
nothing to prove. Otherwise interchange them if needed, so that
a(t) != 0. Then
Ru(s) - Ru(t) = a(s)( u(t) - u(s) ) + ( a(s) - a(t) )( u(q) - u(t) ).
The two terms are bounded by Lip(u) ||s - t||_D and by
(2/ε) ||s - t||_D ε Lip(u) = 2 Lip(u) ||s - t||_D,
respectively. Thus
Lip(Ru) ≤ 3 Lip(u),
which is (5.2b). Then G = u + R gives Lip(G) ≤ 4 Lip(u), which is
(5.2c).
SUPPORT.
For (5.2f)-(5.2g): suppose the new output (Gu or Ru) is nonzero
at s.
* If u(s) != 0, then s|_I is already a point of K_I(u). Its
projection onto K_I(u) has distance 0, hence ≤ ε.
* If u(s) = 0, then the output value at s is a(s) u(q) (for Gu)
or -a(s) u(q) (for Ru), which is nonzero. So u(q) != 0, so
q is in K_I(u), and the condition ||s|_I - q||_∞ < ε gives
dist_∞(s|_I, K_I(u)) < ε.
Either way, the projection s|_I lies in K_I(u)_ε. Taking closures
gives both inclusions (5.2f).
For (5.2g): Ru(s) = 0 whenever ||s|_I - q||_∞ ≥ ε. So the
nonzero set of Ru is contained in the ball closure B_∞(q, ε) in
the I-cube. Taking closures gives (5.2g).
If u = 0 the statements are immediate.∎
5.2 SCALE BUDGETS AND SUPPORT PROPAGATION
For h >= 1, let T_h consist of functions obtained from a single
f_γ, γ = (λ, j, ξ), by m <= h successive operations G or R on
I_γ. A representation with m = 0 is allowed. For m >= 1, the
scales in their order of application must satisfy
ε_1 >= ε_2 >= ... >= ε_m > 0,
sum_{k=1}^{m} ε_k < ℓ_j / 2. (5.3)
The sum is zero for m = 0. Each scale is dyadic and each center
is in its corresponding finite grid.
We refer to γ as a "representing label", since a function may have
more than one representation. The strict inequality in (5.3)
leaves positive budget for a further sufficiently small operation.
The order ensures every operation after a small one has at most
that scale.
Support. For a continuous function u on B, let
supp_B u = closure in B of { s in B : u(s) != 0 }.
All support statements refer to these concrete generated
functions. Later, molecule pairings will ignore additive
constants, so we do not replace a generated function by another
representative modulo constants: that could change its support and
its constant value on a cylinder.
LEMMA 5.2 (Propagation of the correction support). Suppose
u in T_h has a representation based at γ = (λ, j, ξ) with
total scale
e = sum_{k=1}^{m} ε_k.
On supp_B u,
if j >= 1: ||π_j s - ξ||_∞ <= 2 d_j + e,
if j < J_λ: r_λ(s) >= ℓ_j - e. (5.4a)
In particular, on the same support,
if j >= 1: ||π_j s - ξ||_∞ <= 2 d_j + ℓ_j / 2,
if j < J_λ: r_λ(s) >= ℓ_j / 2. (5.4b)
PROOF of Lemma 5.2.
Start with (3.6), which holds on the relative support of f_γ as
well as on its nonzero set, because the two right-hand sides
describe closed subsets of B.
Suppose the claim has been proved for the old function, and add
an operation of scale ε. If the new function is nonzero at s and
the old one is zero there, the proof of Lemma 5.1 gives an old
nonzero value at the zero extension of some grid point q with
||s|_{I_γ} - q||_∞ < ε. Otherwise the old support already
contains s.
Both π_j and r_λ depend only on I_γ. Coordinate restriction has
norm one. And r_λ is the maximum of supremum norms of finitely
many block entries of s, so it is 1-Lipschitz on the cube
[-1, 1]^I. The first upper bound therefore increases by at most
ε, and the second lower bound decreases by at most ε. Induction
proves (5.4a) on the nonzero set. Taking relative closure preserves
the closed inequalities.
Finally, (5.3) gives e < ℓ_j / 2, which yields (5.4b).∎
How these bounds will be used. Lemma 5.2 pins the support of a
generated test u in T_h into a small "annulus" around level
λ = (i, n):
* the first j coordinates are confined to a cylinder of radius
2 d_j + ℓ_j / 2 around ξ (when j >= 1), and
* the local radius r_λ is bounded below by ℓ_j / 2 (when
j < J_λ).
The two alternatives - small j (bounded bands) versus large j
(unbounded bands) - will dominate the case analysis in Section 7.
Bounded bands give a positive local radius threshold. Unbounded
bands concentrate the first j coordinates to a small cylinder.
5.3 TAIL FACTORS AND CLASS BOUNDS
Suppose a test in T_h has support concentrated where the local
radius r_λ is large. Then it lives near one specific coordinate
region. Suppose instead the coordinates concentrate in the tail.
Then the test lives in a region determined by many later
coordinates. Tail factors let us localize independently in either
direction, without disturbing the band structure of Section 5.2.
For N >= 1, define the coordinate tail
τ_N(s) = sup_{k > N} |s_k|,
b_{c,N}(s) = min( 1, ( 2 - τ_N(s)/c )_+ ) (c > 0). (5.5)
Here a_+ = max(a, 0). The function τ_N is 1-Lipschitz in s (with
respect to the supremum norm on D), and
τ_N(s) -> 0 for each s in B as N -> ∞.
Both b_{c,N} and b_{c,N} - 1 have supremum norm at most one, and
Lipschitz constant at most 1/c. Their relative supports satisfy
supp_B b_{c,N} ⊂ { τ_N <= 2c }, (5.6a)
supp_B (b_{c,N} - 1) ⊂ { τ_N >= c }. (5.6b)
VERIFICATION. If τ_N(s) >= 2c, then 2 - τ_N(s)/c <= 0, so the
positive part is 0 and b_{c,N}(s) = 0. That gives (5.6a). If
τ_N(s) <= c, then 2 - τ_N(s)/c >= 1, so the positive part is at
least 1, so b_{c,N}(s) = 1 and b_{c,N}(s) - 1 = 0. That gives
(5.6b). The supremum bounds follow from 0 <= b_{c,N} <= 1. Lipschitz:
τ_N is 1-Lipschitz, τ ↦ (2 - τ/c)_+ is 1/c-Lipschitz, and
min(1, ·) is 1-Lipschitz.
5.4 THE CLASSES V_h
Let V_h consist of all products
u ∏_{r=1}^{l} β_r (5.7)
where
* u in T_h has a representing label with j <= h,
* 0 <= l <= h,
* each β_r is b_{c_r, N_r} or b_{c_r, N_r} - 1, with N_r >= 1
and c_r in { 2^{-a} : 0 <= a <= h }.
The empty product is one. The support of a product is contained
in the support of its cylindrical factor u, so every such test
inherits (5.4b). We will use the countable list of classes
T_1, V_1, T_2, V_2, T_3, V_3, ... (5.8)
The point of alternating T and V in the list is bookkeeping: each
finite level h gives two "test classes" that are considered
distinct in the norm definition of Section 6. This prevents a
hypothetical linear c_0 basis from hiding in one class and
escaping through another.
LEMMA 5.3 (Bounds in each fixed class). With A, L_0 from
(4.21), the following bounds are uniform over all tests of
the indicated class:
||u||_∞ Lip(u)
-----------------------------------------------------------------
u in T_h 2^h A 4^h L_0
u in V_h 2^h A 4^h L_0 + h 4^h A
(5.9)
PROOF of Lemma 5.3.
FOR T_h. Iterating (5.2a)-(5.2e): every application of G keeps
the supremum norm and multiplies the Lipschitz constant by at
most 4; every application of R multiplies the supremum norm by at
most 2 and the Lipschitz constant by at most 3. There are m <= h
operations. The worst case for the supremum norm is m applications
of R, giving 2^h A. The worst case for Lipschitz bounds is h
applications of G, giving 4^h L_0.
FOR V_h. Every tail factor has supremum norm at most one and
Lipschitz constant at most 2^h (since c_r = 2^{-a} with a <= h,
so 1/c_r = 2^a <= 2^h). A product of 0 <= l <= h such factors has
Lipschitz constant at most l * 2^h <= h 2^h, by telescoping the
product difference:
|∏_r β_r(s) - ∏_r β_r(t)|
<= Σ_r (∏_{r' < r} |β_{r'}(t)|) |β_r(s) - β_r(t)|
(∏_{r' > r} |β_{r'}(s)|)
<= Σ_r Lip(β_r) ||s - t||_D
<= l 2^h ||s - t||_D.
The product rule then gives
Lip( u ∏_r β_r )
<= Lip(u) sup(∏_r β_r) + sup(u) Lip(∏_r β_r)
<= 4^h L_0 * 1 + 2^h A * h 2^h
<= 4^h L_0 + h 4^h A.
The supremum norm is at most 2^h A * 1 = 2^h A.∎
CONSEQUENCES. Let us note the key structural features that
Section 7 will use.
* For a fixed bounded set of bands j, the second line of (5.4b)
gives a positive radial threshold r_λ(s) >= ℓ_j / 2 on the
support of any test with band j (and j < J_λ). Because ℓ_j
depends only on j and not on the level λ, this is uniform in λ
for fixed j.
* For j -> ∞, the first line of (5.4b) confines the first j
coordinates to a radius
2 d_j + ℓ_j / 2 -> 0.
So large bands mean the test lives in a thin coordinate
cylinder, shrinking to a point as j grows.
Section 7 will use precisely these alternatives - bounded versus
unbounded bands - to separate or nest the supports of a family of
"detecting" tests.
Section 5 is complete. Sections 6-7 build the Banach space Z from
these tests and prove that Z has no linear copy of c_0.
====================================================================
SECTION 6. MOLECULES AND THE LIFTED CORRECTION
====================================================================
We now turn the selected tests into a Banach space. The norm must
do two jobs:
(i) control point differences so that the correction
h(s) = (S_s - P)s can be lifted Lipschitzly to K: D -> Z;
(ii) remember lists of tests whose signed sums have a common
Lipschitz bound. This is what will rule out a linear c_0
inside Z.
6.1 THE QUOTIENT AND ITS COMPLETION
Let M_0(B) be the real vector space of mass-zero finite formal
sums
m = sum_{s in B} a_s δ_s,
#{ s : a_s != 0 } < ∞, sum_s a_s = 0. (6.1)
For a real function u on B, define the pairing
<u, m> = sum_s a_s u(s).
Constants pair to zero: sum_s a_s = 0 annihilates any constant
function. This follows the mass-zero molecule construction of
Arens and Eells [AE56] and its later Lipschitz-free formulation
[GK03]. The norm below is built from the selected classes (5.8);
no universal property of the usual free space is assumed.
For every triple
d = ( L, C, p ),
where L is one of the listed classes, C in N, and p = 1 + 1/k for
k in N, define
σ_d(m) = sup {
( sum_{ν=1}^{l} | <u_ν, m> |^p )^{1/p} :
l >= 0,
u_1, ..., u_l in L,
Lip( sum_{ν=1}^{l} e_ν u_ν ) <= C
for every (e_ν) in {-1, 1}^l }. (6.2)
The empty list contributes zero. Lists may contain repetitions
and tests with the same representing label; their sole
compatibility condition is the displayed Lipschitz bound for
every choice of signs.
Equivalence of the compatibility formulation. A list is
admissible precisely when
sum_{ν=1}^{l} | u_ν(s) - u_ν(t) | <= C ||s - t||_D
for all s, t in B. (6.3)
Indeed, if (6.3) holds, choose signs matching the increments
u_ν(s) - u_ν(t); the triangle inequality gives
Lip( sum_{ν=1}^{l} e_ν u_ν ) <= C,
for every sign choice. Conversely, if the signed sums all have
Lipschitz constant at most C, choose the signs matching the
increments and apply the definition of Lipschitz.
Now enumerate the triples by d = 1, 2, ..., write C_d for their
integer component, and set
α_d = 2^{-d} / C_d^2,
||m||_0 = ( sum_{d >= 1} α_d σ_d(m)^2 )^{1/2}. (6.4)
The outer square sum and the exponents approaching one serve
distinct purposes in Section 7. The square sum forces one fixed
seminorm to detect a subsequence of any hypothetical c_0 basis.
For that fixed test class and Lipschitz bound, a second exponent
p' < p converts detection by a list into detection by one scalar
test. Both choices are part of the obstruction, as well as of the
norm construction.
LEMMA 6.1 (Exact completed norm). Formula (6.4) is a finite
seminorm on M_0(B). Let N be its kernel, and let Z be the
Banach completion of M_0(B)/N with the induced norm. Then Z
is separable and
ζ : B -> Z, ζ(s) = [ δ_s - δ_0 ] (6.5)
is 1-Lipschitz with ζ(0) = 0. Each σ_d descends and extends
continuously to a seminorm on Z, and the exact formula
||z||_Z^2 = sum_{d >= 1} α_d σ_d(z)^2 (6.6)
holds for every z in Z.
PROOF of Lemma 6.1.
MOLECULE BOUND. For s, t in B and an admissible list for d,
inequality (6.3) bounds the sum of the absolute increments of the
u_ν along the segment from s to t. Since the l_p norm is no larger
than this sum,
σ_d( δ_s - δ_t ) <= C_d ||s - t||_D. (6.7)
FINITENESS. For a fixed admissible list, Minkowski's inequality
makes the expression in (6.2) a seminorm in m; the supremum of
these expressions is also a seminorm, once finite. Every
m = sum_s a_s δ_s of mass zero equals
sum_s a_s ( δ_s - δ_0 ),
because summing the mass-zero gives zero. Therefore
σ_d(m) <= C_d sum_s |a_s| ||s||_D.
This proves finiteness of σ_d, and hence of ||m||_0, because
sum_{d >= 1} α_d C_d^2 = sum_{d >= 1} 2^{-d} = 1.
Minkowski's inequality in l_2, applied to the weighted seminorms,
proves subadditivity of ||.||_0; homogeneity is immediate. Its
kernel is therefore a linear subspace, and quotienting by it
produces a norm.
σ_d IS 1-LIPSCHITZ ON ζ(B). Equation (6.7) and the same square sum
gives
||δ_s - δ_t||_0 <= ||s - t||_D.
This proves ζ is 1-Lipschitz in (6.5) and ζ(0) = 0.
SEPARABILITY. The ball B is separable because D is a
countable-coordinate c_0 space. If B_0 is a countable dense
subset, continuity of ζ makes ζ(B_0) dense in ζ(B). The real
linear span of ζ(B) is the quotient molecule space; its rational
span over B_0 is consequently dense in Z. Hence Z is separable.
σ_d DESCENDS AND EXTENDS. For molecules m, m':
|σ_d(m) - σ_d(m')| <= σ_d(m - m')
<= α_d^{-1/2} ||m - m'||_0. (6.8)
In particular σ_d vanishes on N, descends, and extends
continuously to Z. Subadditivity and homogeneity pass to limits.
EXACT FORMULA (6.6). The map
m |-> ( √α_d σ_d(m) )_{d >= 1}
from M_0(B) into l_2 is 1-Lipschitz, because the coordinate
differences are bounded by √α_d σ_d(m - m'), and the l_2 norm is
at most the sum of the coordinates' magnitudes. If molecules
converge to z in Z, their images therefore converge in l_2. The
d-th coordinate of the limit is √α_d σ_d(z) by continuity of each
seminorm. The norms of these images are exactly the molecule
norms, which converge to ||z||_Z. Taking squares gives (6.6).
This proves Lemma 6.1.∎
6.2 FIXED-CLASS DUALITY
The obstruction argument of Section 7 will modify a test through a
pointwise limit after a vector in Z has already been fixed. The
next lemma justifies that passage, and also retains the exact list
formula (6.2) after completion.
LEMMA 6.2 (Dual tests and pointwise limits). Every test in
T_h or V_h defines a bounded linear functional on Z by its
molecule pairing. For each fixed class L, there is a finite
common bound B_L for these dual norms.
For each fixed triple d = (L, C, p), formula (6.2) remains
exact on Z, with the extended test pairings.
If u_n in L and u_n(s) -> u(s) for every s in B, then the
pointwise limit defines a bounded functional on Z of norm at
most B_L, and <u_n, z> -> <u, z> for each fixed z in Z. More
precisely, for any molecule class m and any z in Z,
| <u_n - u, z> |
<= | <u_n - u, m> | + 2 B_L ||z - m||_Z. (6.9)
PROOF of Lemma 6.2.
BOUNDED FUNCTIONAL ON Z. Choose an integer C_L no smaller than
the uniform Lipschitz bound in Lemma 5.3, so
Lip(u) <= C_L for every u in L.
Every singleton (u), u in L, is admissible for the one triple
d_L = ( L, C_L, 2 ),
because with l = 1 a signed sum is just ±u, whose Lipschitz
constant is at most C_L. Thus
| <u, m> | <= σ_{d_L}(m) <= α_{d_L}^{-1/2} ||m||_0. (6.10)
The pairing vanishes on N (because N has zero molecule norm and
σ_{d_L} does) and extends to Z^*. Let
B_L = α_{d_L}^{-1/2}.
Then ||u||_{Z^*} <= B_L for every u in L, uniformly.
EXTENSION OF σ_d TO Z. For an admissible list
u = ( u_1, ..., u_l )
of a fixed triple d, its evaluation map
Λ_u m = ( <u_ν, m> )_{ν=1}^l in l_p^l
has operator norm at most α_d^{-1/2} on the molecule quotient, by
the same argument as (6.10) applied to the list formula (6.2).
Therefore Λ_u extends with that bound to Z.
The supremum of || Λ_u z ||_p over all admissible lists is finite
and α_d^{-1/2}-Lipschitz as a seminorm in z: its change between
z, z' is bounded by the same supremum at z - z'. This supremum
agrees with σ_d on the dense molecule quotient. It is therefore
the continuous extension of σ_d, which proves the exact formula
(6.2) on Z with the extended pairings.
POINTWISE LIMITS. For a pointwise-convergent sequence u_n of
tests in one class L, the pairings converge on every finite
molecule:
<u_n, m> = sum_s a_s u_n(s) -> sum_s a_s u(s) = <u, m>,
since m has finite support and u_n(s) -> u(s) pointwise.
Inequality (6.10) passes to the pointwise limit, so the molecule
functional u extends with norm at most B_L.
To prove (6.9), decompose:
| <u_n - u, z> |
= | <u_n - u, z - m > + <u_n - u, m > |
<= | <u_n - u, z - m > | + | <u_n - u, m > |.
The first term is at most
( ||u_n||_{Z^*} + ||u||_{Z^*} ) || z - m ||_Z
<= 2 B_L || z - m ||_Z,
by (6.10) for ||u_n||_{Z^*} and by the pointwise-limit passage
for ||u||_{Z^*}. This gives (6.9).
CONVERGENCE ON EACH FIXED z. Let eps > 0 be prescribed. Choose a
molecule class m with
|| z - m ||_Z < eps / ( 4 B_L )
(such an m exists because the molecule quotient is dense in Z).
Then, because m has finite support,
< u_n - u, m > -> 0 as n -> ∞.
So choose n large enough that | <u_n - u, m> | < eps / 2. Then
by (6.9),
| <u_n - u, z> | < eps / 2 + 2 B_L * eps / (4 B_L) = eps.
This proves convergence on z.
The choice of m and n may depend on z; no convergence uniform over
a moving sequence of vectors is asserted. Only pointwise (that is,
fixed-vector) convergence is used. This completes the proof of
Lemma 6.2.∎
6.3 THE c_0-VALUED CORRECTION MAP
LEMMA 6.3 (The c_0-valued correction map). There is a bounded
linear map Q: Z -> U such that
Q ζ(s) = h(s) = ( S_s - P ) s (s in B). (6.11)
PROOF of Lemma 6.3.
Define on formal molecules
Q_0 m = sum_s a_s h(s) in U.
This finite sum lies in U, not merely in l_infinity, because each
h(s) lies in U.
Each coordinate function f_γ belongs to T_1 through its zero-
operation representation. Choose an integer C_0 >= L_0 and the
triple
d_0 = ( T_1, C_0, 2 ).
Every singleton (f_γ) is admissible (since f_γ in T_1 and
Lip(f_γ) <= L_0 <= C_0), whence
|| Q_0 m ||_U = sup_γ | < f_γ, m > |
<= σ_{d_0}(m)
<= α_{d_0}^{-1/2} ||m||_0.
Thus Q_0 vanishes on N, descends, and extends to a bounded linear
map into the complete space U.
Since h(0) = (S_0 - P)·0 = 0, applying Q to the elementary molecule
δ_s - δ_0 gives
Q ζ(s) = Q( δ_s - δ_0 ) = Q_0( δ_s - δ_0 )
= h(s) - h(0) = h(s).
This proves (6.11).∎
6.4 RADIAL EXTENSION
LEMMA 6.4 (Lipschitz lift). Define
K(0) = 0,
K(x) = ||x||_D * ζ( x / ||x||_D ) (x != 0). (6.12)
Then K: D -> Z is positively homogeneous and
Lip(K) <= 3.
It satisfies the exact identity
Q K = g - P. (6.13)
PROOF of Lemma 6.4.
BOUND ON ζ. Since ζ(0) = 0 and Lip(ζ) <= 1 (Lemma 6.1),
|| ζ(s) ||_Z <= 1 for every s in B.
LIPSCHITZ BOUND. Take x, x' in D, both nonzero. Put
a = ||x||_D, b = ||x'||_D,
s = x / a, s' = x' / b.
Without loss of generality a >= b. Splitting K(x) - K(x') at
b ζ(s):
|| K(x) - K(x') ||_Z
= || a ζ(s) - b ζ(s') ||_Z
<= ( a - b ) || ζ(s) ||_Z + b || ζ(s) - ζ(s') ||_Z
<= || x - x' ||_D + b || s - s' ||_D.
The normalization estimate (4.7) with M = a >= b bounds the last
term by 2 ||x - x'||_D. Hence
|| K(x) - K(x') ||_Z <= 3 || x - x' ||_D.
If one endpoint is zero, use || K(x) ||_Z <= ||x||_D, which is
even stronger (Lipschitz constant 1 on the endpoint). This proves
Lip(K) <= 3.
POSITIVE HOMOGENEITY. For λ > 0 and x != 0,
K(λx) = ||λx||_D ζ( λx / ||λx||_D )
= λ ||x||_D ζ( x / ||x||_D )
= λ K(x).
If x = 0, both sides are 0.
THE IDENTITY (6.13). For x != 0,
Q K(x) = ||x||_D Q ζ( x / ||x||_D )
= ||x||_D h( x / ||x||_D ).
By the shift-correction identity (4.5), the right-hand side is
exactly (g - P)(x). If x = 0, both sides vanish. Therefore (6.13)
holds for every x in D.∎
What we have. The three inputs of Lemma 2.1 are now present:
* g: D -> U, onto bi-Lipschitz (Propositions 4.2 and 4.3);
* Q: Z -> U, bounded linear (Lemma 6.3);
* K: D -> Z, Lipschitz with QK = g - P (Lemma 6.4).
No surjectivity of Q was used or proved. Lemma 2.1 will complete
the construction once we know Z has no linear copy of c_0. That
is the task of Section 7.
====================================================================
SECTION 7. DETECTORS EXCLUDE LINEAR c_0
====================================================================
A linear copy of c_0 in Z would provide vectors bounded away from
zero whose finite signed sums are uniformly bounded. We first show
that one class of tests detects a subsequence of these vectors. We
then identify the additional property needed for a contradiction:
the tests' finite signed sums must have one Lipschitz bound. The
rest of the section obtains this property by modifying the
detectors and separating or nesting their supports. Supports
always refer to the concrete generated functions on B.
7.1 FROM BOUNDED SIGNED SUMS TO INDIVIDUAL DETECTORS
The extraction has two stages. The outer square sum prevents
detection from escaping through many different seminorms; within
one seminorm, the exponents approaching one prevent it from
spreading over many small scalar pairings.
LEMMA 7.1 (One class detects a subsequence). Suppose (m_i) in
Z satisfies, for some r, M > 0,
||m_i||_Z >= r,
|| sum_{i in H} e_i m_i ||_Z <= M (7.1)
for every finite H ⊂ N and every sign choice e_i in {-1, 1}.
Then u(m_i) -> 0 for every u in Z^*. After passing to a
subsequence, there are tests F_i in one listed class L and a
constant δ > 0 with
| < F_i, m_i > | >= δ. (7.2)
PROOF of Lemma 7.1.
WEAK NULLITY. For any fixed u in Z^*, choosing signs matching
u(m_i) in (7.1) gives
sum_{i in H} | u(m_i) | <= M ||u||_{Z^*} for every finite H.
Therefore sum_i |u(m_i)| < ∞, in particular u(m_i) -> 0. Also
||m_i||_Z <= M by choosing H = {i} and either sign.
ONE FIXED SEMINORM STAYS AWAY FROM ZERO. Suppose, for
contradiction, that σ_d(m_i) -> 0 for every d. Then the vectors
v_i = ( √α_d σ_d(m_i) )_{d >= 1}
converge coordinatewise to 0 in l_2, while
r <= ||v_i||_2 = ||m_i||_Z <= M.
GLIDING HUMP EXTRACTION. A direct gliding-hump selection gives
disjoint finite intervals A_i of test indices, after a subsequence,
with
sum_{d in A_i} α_d σ_d(m_i)^2 >= r^2 / 4. (7.3)
The idea: once the endpoint of the preceding interval is fixed,
coordinatewise convergence makes the squared mass before that
endpoint < r^2/4 for sufficiently large i. Choose a new finite
endpoint so that the squared tail mass is also < r^2/4. The mass
in between is >= r^2/2, which implies (7.3).
TWO-VARIABLE JENSEN. The function σ_d^2 is even and convex, because
σ_d is a seminorm and squaring is convex and increasing on
[0, ∞). For fixed y, m,
( σ_d(y+m)^2 + σ_d(y-m)^2 ) / 2 >= σ_d(m)^2. (7.4)
For independent random signs on a finite set H ⊂ N and a
distinguished index i in H, average (7.4) over e_i with all other
signs fixed. Then average over remaining signs. This yields
E σ_d( sum_{k in H} e_k m_k )^2 >= σ_d(m_i)^2,
the expectation over the uniform distribution on signals in
{-1, 1}^H. So for each fixed d,
E σ_d( sum_{k in H} e_k m_k )^2 >= σ_d(m_i)^2
for every i in H.
ASSEMBLING. The inequality (7.1) already holds for every sign
choice, so we average the squared norm over all 2^|H| such
choices. The exact norm formula (6.6), the independence of the
signs e_k, and the disjointness of the blocks A_i give
M^2 >= E || sum_{k in H} e_k m_k ||_Z^2
= sum_d α_d E σ_d( sum_{k in H} e_k m_k )^2
>= sum_{i in H} sum_{d in A_i}
α_d σ_d(m_i)^2
>= sum_{i in H} r^2 / 4
= |H| r^2 / 4.
For H large enough this contradicts ||S||_Z <= M. Hence for some
d = (L, C, p), we have
σ_d(m_i) >= a > 0 on a subsequence.
EXPONENT DOWNGRADE. Choose the listed exponent p' = 1 + 1/k' < p
with the same class L and integer C, and call the triple d'. Its
seminorm is bounded on these vectors by
σ_{d'}(m_i) <= α_{d'}^{-1/2} M = B_0.
For each i, choose a finite admissible list for d whose l_p value
on m_i is at least a/2. Put x_ν = |<u_ν, m_i>|. The same list is
admissible for d' because only the exponent changed. Therefore
(a/2)^p <= sum_ν x_ν^p
<= ( max_ν x_ν )^{p - p'} sum_ν x_ν^{p'}
<= ( max_ν x_ν )^{p - p'} B_0^{p'}.
The list is nonempty and B_0 > 0. One of its tests therefore has
evaluation at least
δ = ( (a/2)^p / B_0^{p'} )^{1/(p - p')} > 0.
Choosing that test as F_i proves (7.2).∎
7.2 GROWTH FROM COMPATIBLE DETECTORS
Call a sequence of tests "compatible" if all its finite signed sums
have a common Lipschitz bound. The norm σ_d was defined using
precisely such lists. Once compatible tests detect the vectors,
a single seminorm forces the following growth.
LEMMA 7.2 (Growth from compatible detectors). Let (m_i) be a
sequence in Z, and let F_i belong to one listed class L.
Suppose there are δ > 0 and an integer C >= 1 such that
| < F_i, m_i > | >= δ for every i,
Lip( sum_{i in H} e_i F_i ) <= C for every i,
finite H, signs e_i.
For the listed triple d = (L, C, 2), every nonempty finite H
satisfies
max_{e_i in {-1, 1}} || sum_{i in H} e_i m_i ||_Z
>= √α_d · δ · √|H|. (7.5)
PROOF of Lemma 7.2.
The list ( F_k )_{k in H} is admissible for the triple
d = (L, C, 2): all its finite signed sums have Lipschitz constant
at most C. By Lemma 6.2, the exact list formula (6.2) extends to
Z, so the admissible list (F_k)_{k in H} gives the lower bound
σ_d( S ) >= ( sum_{k in H} | <F_k, S> |^2 )^{1/2},
where S = sum_{i in H} e_i m_i is the current random signed sum.
For independent random signs e_i in {-1, 1}:
<F_k, S> = sum_{i in H} e_i <F_k, m_i>,
| <F_k, S> |^2 = sum_{i, j in H} e_i e_j <F_k, m_i> <F_k, m_j>.
Taking the expectation over the 2^{|H|} sign choices, cross terms
with i != j cancel because E[e_i e_j] = 0 for i != j, and the
diagonal terms give E[e_i^2] = 1. Thus
E | <F_k, S> |^2 = sum_{i in H} | <F_k, m_i> |^2
>= | <F_k, m_k> |^2
>= δ^2.
Summing over k in H,
E σ_d( S )^2
>= sum_{k in H} E | <F_k, S> |^2
>= sum_{k in H} δ^2
= |H| δ^2.
Combining with the exact norm formula (6.6),
E || S ||_Z^2
= sum_{d' >= 1} α_{d'} E σ_{d'}( S )^2
>= α_d E σ_d( S )^2
>= α_d |H| δ^2.
Since the maximum over sign choices is at least the average over
the 2^{|H|} choices,
max_{e_i in {-1, 1}} || sum_{i in H} e_i m_i ||_Z^2
>= α_d |H| δ^2.
Taking square roots gives (7.5).∎
7.3 SUPPORT CONDITIONS FOR COMPATIBILITY
Localizing detecting Lipschitz functions and combining them with a
common Lipschitz bound is also central to Kalton's arguments for
metric free spaces [Kalton04, Lemma 4.5 and proof of Theorem 4.6].
Here the supports are finite-coordinate cylinders and tail regions,
and their compatibility must be retained within the selected test
classes defining Z.
LEMMA 7.3 (Compatible supports). Let B be a convex subset of a
normed space. Let u_i, b_i be continuous real functions on B,
with b_i Lipschitz, and put F_i = u_i b_i. Suppose
Lip(F_i) <= L for every i.
(1) If the closed relative supports supp_B F_i are pairwise
disjoint, every finite signed sum of the F_i has Lipschitz
constant at most L.
(2) Suppose that for each i there is a relatively open set
C_i ⊂ B and a scalar e_i such that u_i = e_i on C_i and
supp_B u_k ⊂ C_i for every k > i. If
sum_i | e_i | Lip(b_i) < ∞,
every finite signed sum of the F_i has Lipschitz constant
at most
L + sum_i | e_i | Lip(b_i).
If b_i = 1 for every i, the summability condition is
automatic.
PROOF of Lemma 7.3.
CASE (1): DISJOINT SUPPORTS. Fix a finite set of indices H and
signs e_i in {-1, 1}. At a given point s in B, at most one of
the closed supports supp_B F_i (for i in H) contains s, by
pairwise disjointness. All other summands of
G = sum_{i in H} e_i F_i
vanish on a relative neighborhood of s. So G is locally either
zero or a single signed F_i, with Lipschitz constant at most L.
To globalize: take s, t in B. Convexity of B makes the segment
[s, t] contained in B. On this segment, G is locally equal to
one of the finitely many signed F_i or zero, hence locally
Lipschitz with constant L. A Lebesgue-number subdivision of [s,t]
into pieces on which the local form is constant then gives
| G(s) - G(t) | <= L ||s - t||_D,
because the lengths of the collinear sub-segments add to
||s - t||_D. This proves Lip(G) <= L.
CASE (2): NESTED PLATEAUS. Fix a finite H and signs e_i in
{-1, 1}. Consider
G = sum_{i in H} e_i F_i = sum_{i in H} e_i u_i b_i.
Take a point s in B. If no support supp_B u_j (over selected
j in H) contains s, then every u_j vanishes on a common
neighborhood of s (there are only finitely many closed supports
and s is outside all of them). So G vanishes on that
neighborhood, giving local Lipschitz constant 0 <= L.
Otherwise, let j be the largest selected index whose u_j-support
contains s. Later selected u_k (k > j) vanish on a neighborhood
of s, because their own u_k-supports are closed and miss s.
Earlier selected u_i (i < j) equal e_i on a neighborhood of s:
indeed the chosen j has s in supp_B u_j, and the nesting hypothesis
puts supp_B u_j ⊂ C_i for every i < j, so s in C_i, so u_i = e_i
on some relative neighborhood of s inside C_i.
Therefore, on a small enough neighborhood of s,
G = sum_{i < j, i selected} e_i e_j b_i (with matching signs)
+ e_j F_j
+ 0.
More precisely, the signed sum equals
sum_{i < j, i selected} (e_i * e_i) b_i + e_j F_j,
where the first term is a (finite) signed sum of the b_i alone.
Its Lipschitz constant on the neighborhood is at most
sum_{i < j} | e_i | Lip(b_i). The second term e_j F_j has
Lipschitz constant at most L by hypothesis. So the local Lipschitz
bound on G near s is
L + sum_{i in H} | e_i | Lip(b_i).
The same convexity-plus-Lebesgue-number subdivision as in case (1)
upgrades local bounds to
Lip(G) <= L + sum_i | e_i | Lip(b_i).
When b_i = 1 for all i, Lip(b_i) = 0 and the summability condition
is automatic, giving Lip(G) <= L.∎
7.4 TAIL SEPARATION
LEMMA 7.4 (Tail separation). Suppose F_i belong to one listed
class and have representations with bounded band indices.
Suppose m_i in Z and
| <F_i, m_i> | >= δ > 0,
supp_B F_i ⊂ { τ_{M_i} >= c_* }, M_i in N, M_i -> ∞,
c_* > 0. (7.7)
Then a subsequence has modified tests in one class V_H, with
detection at least δ/2, whose relative supports are pairwise
disjoint.
PROOF of Lemma 7.4.
Choose a dyadic c = 2^{-a} with 2c < c_*. If the original class
is T_h, view a test as a cylindrical factor with no tail factors;
if it is V_h, keep its chosen product representation. A single
integer H, larger than h, the bounded band indices, and a, can
be chosen so that both F_i and F_i b_{c, N} belong to V_H for all
i, N. There is room for one extra tail factor.
For each fixed i, b_{c, N}(s) -> 1 at every s in B, because
τ_N(s) -> 0. Lemma 6.2 (used in this single class V_H) therefore
gives
<F_i b_{c, N}, m_i> -> <F_i, m_i> as N -> ∞.
Choose N large enough that the modulus of the difference is at
most δ/2. Every sufficiently large N then works. This choice is
made for the fixed vector m_i.
Select indices recursively. After choosing an index i and such
an N = N_i, take the next original index so far out that its
M_k exceeds all previously selected N_i (this is possible because
M_k -> ∞). Then choose its own N = N_k as above. Relabel the
selected indices 1 <= i_1 < i_2 < ... .
For selected k > i, the new i-th support requires τ_{N_i} <= 2c
by (5.6a). The new k-th support requires τ_{M_k} >= c_* by (7.7),
hence
τ_{N_i} >= τ_{M_k} >= c_* > 2c,
because N_i < M_k and τ_N(s) = sup_{j > N} |s_j| is nonincreasing
in N. This contradicts the i-th support condition τ_{N_i} <= 2c
on the intersection of the two supports. So the selected supports
are pairwise disjoint.
The detection bound follows from the choice of N_i: on m_i, the
modified test F_i b_{c, N_i} has pairing at least
δ - δ/2 = δ/2.∎
7.5 SELECTING COMPATIBLE DETECTORS
PROPOSITION 7.5 (Selection of compatible detectors). Let
(m_i) in Z be weakly null, meaning u(m_i) -> 0 for every
u in Z^*. Suppose F_i belong to one listed class L and
| < F_i, m_i > | >= δ > 0 (i >= 1).
There are a subsequence (m_{i_k}), a listed class L_1, tests
F̃_k in L_1, a constant δ_1 > 0, and an integer C_1 >= 1 such
that
| < F̃_k, m_{i_k} > | >= δ_1,
Lip( sum_{k in H} e_k F̃_k ) <= C_1
for every k, every finite H, and all signs e_k.
PROOF of Proposition 7.5.
We pass to subsequences and relabel their indices throughout the
proof. Every fixed test pairs to zero in the limit, by weak
nullity and Lemma 6.2.
Fix a concrete representation of each detector F_i. If its class
is T_h, take F_i = u_i (no tail factor). If its class is V_h,
write F_i = u_i b_i where u_i is its chosen cylindrical factor
and b_i is its product of tail factors. Let
γ_i = ( λ_i, j_i, ξ_i )
be the chosen label.
After subsequences, either the labels are pairwise distinct or
one single label is repeated throughout. In the distinct case,
the bands have a bounded subsequence or a subsequence tending to
infinity. We handle these three possibilities separately.
CASE 1: DISTINCT LABELS WITH BOUNDED BANDS.
Every block Γ_λ is finite, so pairwise distinct labels force
λ_i to leave every finite set. Since
{ λ : J_λ <= J }
is finite for every J, we have J_{λ_i} -> ∞. Pass to a further
subsequence with one fixed band j. Eventually j < J_{λ_i}, and
(5.4b) gives
r_{λ_i} >= c_* := ℓ_j / 2 > 0
on supp_B F_i.
The coordinates used by these radii escape every fixed initial
segment. Indeed, each coordinate occurs in at most two radii, so
a finite set of coordinates occurs in only finitely many levels.
After discarding finitely many terms, choose integers M_i -> ∞
so that every coordinate used by r_{λ_i} has index > M_i. Then
r_{λ_i}(s) <= τ_{M_i}(s) for every s in B.
The condition r_{λ_i} >= c_* is the hypothesis of Lemma 7.4, so
we get detectors in one enlarged class with pairwise disjoint
relative supports. Lemma 7.3(1) gives a common Lipschitz bound
for all finite signed sums. Set C_1 equal to this bound.
CASE 2: DISTINCT LABELS WITH BANDS TENDING TO INFINITY.
Now j_i -> ∞. The fixed class must be T_h, because a V_h
representation has j <= h.
DIAGONAL EXTRACTION. By diagonal extraction from the compact
intervals [-1, 1], the finite tags
ξ_i ∈ [-1, 1]^{j_i}
converge to a single vector
v_* ∈ [-1, 1]^N
coordinate by coordinate. The point v_* need not be in c_0;
only its finite restrictions will be used.
Write
a_i = 2 d_{j_i} + ℓ_{j_i} / 2 -> 0.
For every fixed nonempty finite coordinate set I, once
j_i >= max I, inequality (5.4b) gives
sup_{s in supp_B F_i} || s|_I - v_*|_I ||_∞
<= a_i + || ξ_i|_I - v_*|_I ||_∞ -> 0. (7.14)
CYLINDER THE TEST. For each fixed i, consider arbitrarily small
positive dyadic θ, chosen below the last existing scale if there
is one and with θ less than the remaining strict budget in (5.3).
Choose a grid center q_i(θ) within θ/8 of v_*|_{I_{γ_i}}. The
test
F_{i,θ} = G_{q_i(θ), θ} F_i
belongs to T_{h+1}, as does F_i, and
|| F_{i,θ} - F_i ||_∞ <= θ Lip(F_i) <= θ 4^h L_0.
So F_{i,θ} -> F_i pointwise as θ ↓ 0, even though the centers
move. For the fixed i, Lemma 6.2 and (6.9) allow a choice θ_i
with
| < F_{i, θ_i} - F_i, m_i > | < δ / 2.
This is a separate choice after fixing m_i, not a uniform limit
over the sequence of vectors. Write F̃_i = F_{i, θ_i}. It detects
m_i with bound δ/2.
PLATEAU ON A CYLINDER. The function F̃_i is constant on the
relatively open cylinder
C_i = { s in B :
|| s|_{I_{γ_i}} - v_*|_{I_{γ_i}} ||_∞ < 3 θ_i / 8 } (7.15)
because the distance from its grid center is then less than
θ_i / 2. Its enlarged representation still satisfies the strict
budget, so (5.4b) applies to F̃_i. Therefore (7.14) holds with
F̃_i in place of F_i. For each fixed earlier i, all sufficiently
late supports lie in C_i. A recursive subsequence makes this
true for every later selected index simultaneously.
NESTED PLATEAUS. Now Lemma 7.3(2), with u_i = F̃_i and
b_i = 1 (and hence Lip(b_i) = 0), gives a common Lipschitz bound
for finite signed sums. The bound is the uniform Lip(F̃_i) bound
from Lemma 5.3 applied to class T_{h+1}:
Lip( sum_{k in H} e_k F̃_k ) <= 4^{h+1} L_0.
Set C_1 equal to this constant.
CASE 3: ONE REPEATED LABEL.
Now a single label γ = (λ, j, ξ) is repeated. All cylindrical
factors depend on the same finite set I = I_γ. We first expand
the tests to isolate a varying term that still detects the
vectors. An escaping tail index then gives disjoint supports.
Otherwise, a first small remainder R gives shrinking cylindrical
supports; a final local modification produces nested plateaus
with summably small values.
EXTRACT A SUBSEQUENCE. Fix the number m <= h and the types
(G or R) of the cylindrical operations, as well as the number
l <= h and the types of the tail factors. For each operation
position, its dyadic scale has a subsequence that is either
constant or tending to zero. In the constant case, its center
can also be made constant, because the permitted grid in the
fixed cube is finite. At each tail position, the scale c can be
made constant, and its integer index N is either constant or
tending to infinity. Make all these finitely many extractions
together.
EXPANSION AND DETECTION. Call an ingredient "fixed" when all of
its parameters are now constant. The parameters in each concrete
representation are frozen before the algebraic expansion.
Replace every nonfixed G_{q,ε} by Id + R_{q,ε}, and every
nonfixed factor b_{c,N} by 1 + (b_{c,N} - 1). The operations are
linear in their input functions, so these identities distribute
through later operations with no parameters recomputed. The
expansion is only of the cylindrical operators and of the tail
product separately; it never applies G to a tail-dependent full
test.
There are at most K_0 = 2^{m+l} <= 2^{2h} resulting term
positions. Each cylindrical term retains a chronological
subsequence of the original operations, possibly replacing G by
R at the same scale. Its scales remain nonincreasing and its
total scale does not increase. Omitting a tail factor or
replacing b by b - 1 also preserves the permitted type and
scale. Because j is fixed, all resulting full terms belong to
one class V_H for some fixed H >= max{h, j}, even if the
original class was T_h.
Write the exact expansion
F_i = sum_{a=1}^{K} F_i^a, K <= K_0. (7.23)
A term position whose only ingredients are fixed is a fixed
concrete test F^a. Its pairing with m_i tends to zero by weak
nullity; it is not asserted to vanish identically. There are only
finitely many such positions, so their total pairing tends to
zero. In view of the detection |<F_i, m_i>| >= δ, for all
sufficiently large i, the sum of the absolute pairings of the
varying terms in (7.23) is at least δ/2. One of the at most K_0
positions has pairing at least δ/(2K_0); a subsequence fixes that
position. Relabel its tests as
F_i = u_i b_i, | < F_i, m_i > | >= δ_0 > 0. (7.24)
ESCAPING TAIL INDEX. If b_i contains a nonfixed factor
b_{c, N_i} - 1 with N_i -> ∞, its support is contained in
{ τ_{N_i} >= c } by (5.6b). The band is fixed, so Lemma 7.4
applies. We may therefore assume no such factor remains. Every
retained nonfixed tail factor would have this form after
expansion. The remaining tail product is therefore fixed, and
the variation in (7.24) must occur in the cylindrical factor.
Every retained nonfixed cylindrical operation is an R: a
nonfixed G was replaced by an omitted identity or by R.
A SMALL REMAINDER. Choose the first retained R whose scale
ε_i -> 0, and call its center q_i. All retained operations
before it are fixed, so they produce one fixed prefix function
p from f_γ. Set
L_pre = Lip(p) <= 4^h L_0.
This is the Lipschitz bound of the prefix, not the base bound
L_0 itself; earlier operations may have enlarged it. If
L_pre = 0, this R and the resulting test vanish, contradicting
(7.24). Lemma 5.1 gives
|| R_{q_i, ε_i} p ||_∞ <= L_pre ε_i,
K_I( R_{q_i, ε_i} p ) ⊂ closure B_∞( q_i, ε_i ).
Every retained later scale is at most ε_i, by the nonincreasing
order in the original representation. A later G does not
increase the supremum norm; a later R increases it by at most a
factor two. Either enlarges projected support by at most its
scale. Iterating the two parts of Lemma 5.1 over at most h later
operations yields
|| u_i ||_∞ <= 2^h L_pre ε_i,
{ s|_I : s in supp_B u_i } ⊂ K_I(u_i)
⊂ closure B_∞( q_i, (h + 1) ε_i ). (7.25)
The first inclusion follows by continuity of coordinate
restriction and the definition of relative support. This is where
the order of scales is essential: a fixed total budget alone
would not make the sum of the later scales tend to zero.
PLATEAU. Pass to a subsequence with q_i -> v in the compact cube
[-1, 1]^I. We now make each cylindrical factor constant on a
small cylinder about v, while preserving (7.24). For a fixed i,
choose arbitrarily small positive dyadic θ that are below the
last retained scale, below the remaining strict budget, and at
most ε_i. Let q_i^+(θ) be a permitted grid point within θ/8 of
v, and set
F_{i, θ} = ( G_{q_i^+(θ), θ} u_i ) b_i.
All these full tests, and F_i = u_i b_i, belong to one fixed
enlarged class V_{H'} with H' large enough for one more
cylindrical operation. The tail product has supremum norm at
most one, so
|| F_{i, θ} - F_i ||_∞ <= θ Lip(u_i).
For the fixed vector m_i, pointwise convergence and (6.9) give
an allowed choice θ_i such that
| < F_{i, θ_i} - F_i, m_i > | < δ_0 / 2. (7.26)
Again the scale is chosen separately for each m_i; no uniform
dual convergence on the moving sequence is used.
Write u_i := G_{q_i^+(θ_i), θ_i} u_i and F_i := u_i b_i.
This F_i detects m_i with bound δ_0 / 2. The support enlargement
in Lemma 5.1, (7.25), and θ_i <= ε_i give
{ s|_I : s in supp_B u_i } ⊂ closure B_∞( q_i, (h + 2) ε_i ). (7.27)
This conclusion does not require the new grid center to be close
to q_i: a new nonzero value can be created only near an old
nonzero value, which is exactly the support enlargement proved
earlier. As q_i -> v and ε_i -> 0, these cylindrical supports
concentrate at v.
PLATEAU VALUES. On the relatively open cylinder
C_i = { s in B : || s|_I - v ||_∞ < 3 θ_i / 8 },
the function u_i is the constant
e_i = u_i( q_i^+(θ_i) ).
Equation (7.25) gives
| e_i | <= 2^h L_pre ε_i -> 0. (7.28)
NESTED PLATEAUS. Choose a recursive subsequence on which
sum_i | e_i | < ∞,
and
supp_B u_k ⊂ C_i whenever k > i.
To see that both requirements can be imposed: after selecting
finitely many indices, use (7.27) to discard a finite initial
segment so that all remaining supports lie in their cylinders;
then choose the next index with, for example, | e_i | <= 2^{-i},
using (7.28). Continue.
FINISHING THE COMPATIBILITY. The remaining b_i are products of
at most h tail factors of supremum norm at most one and Lipschitz
constant at most 2^h. Thus Lip(b_i) <= h 2^h uniformly. The full
F_i have a common Lipschitz bound because they are in V_{H'}, and
sum_i | e_i | Lip(b_i) < ∞.
Lemma 7.3(2), applied to the cylindrical supports in (7.27),
gives the required common Lipschitz bound for all finite signed
sums. The nesting is of supp_B u_i, not merely the possibly
smaller supports of F_i.
In each case the modified tests remain in one listed class,
retain a positive common lower bound on their pairings, and have
a common Lipschitz bound for finite signed sums. Choose an
integer C_1 no smaller than this last bound. This completes the
proof of Proposition 7.5.∎
7.6 EXCLUDING A LINEAR COPY OF c_0
PROPOSITION 7.6. The Banach space Z contains no linearly
isomorphic copy of c_0.
PROOF of Proposition 7.6.
Suppose T: c_0 -> Z is an isomorphic embedding, with constants
0 < a <= b < +∞ such that
a || x ||_{c_0} <= || T x ||_Z <= b || x ||_{c_0}
for every x in c_0.
Put m_i = T e_i, where e_i is the i-th standard basis vector of
c_0.
BOUNDED SIGNED SUMS. For any nonempty finite H ⊂ N and any signs
e_i in {-1, 1},
x = sum_{i in H} e_i e_i satisfies || x ||_{c_0} = 1.
Indeed x has entries in {-1, 0, 1}, with at least one entry equal
to ±1, so its supremum is 1. Therefore
a = a * 1 <= || T x ||_Z = || sum_{i in H} e_i m_i ||_Z
<= b * 1 = b.
Also || m_i ||_Z = || T e_i ||_Z >= a || e_i ||_{c_0} = a. So the
hypothesis (7.1) of Lemma 7.1 holds with r = a, M = b.
Lemma 7.1 gives a weakly null subsequence (m_{i_k}) and detectors
F_k in one listed class L_1, with | < F_k, m_{i_k} > | >= δ > 0
for all k.
Proposition 7.5 then gives a further subsequence, a listed class
L_2, compatible detectors F_k in L_2 with a common nonzero pairing
δ_1 > 0, and an integer C_1 >= 1 such that
| < F_k, m_{i_k} > | >= δ_1,
Lip( sum_{k in H} e_k F_k ) <= C_1
for every k, every finite H, and all signs e_k.
Apply Lemma 7.2 to this compatible family, with L = L_2, δ = δ_1,
C = C_1, and d_1 = (L_2, C_1, 2). The conclusion (7.5) gives
max_{e in {-1, 1}^H} || sum_{k in H} e_k m_{i_k} ||_Z
>= √α_{d_1} · δ_1 · √|H|
for every nonempty finite set H of selected indices. But the
left-hand side is at most b by the bounded-signed-sum bound
above. Therefore
b >= √α_{d_1} · δ_1 · √|H| for every finite H.
Since α_{d_1} > 0 and δ_1 > 0, the right-hand side grows without
bound as |H| grows. This is a contradiction. Hence no such
embedding T exists.∎
====================================================================
SECTION 8. ASSEMBLY AND SAME-SPACE CONSEQUENCES
====================================================================
We now collect everything to prove Theorem 1.1 and its three
same-space consequences.
PROOF of Theorem 1.1.
Lemma 6.1 gives a separable real Banach space Z. Proposition 7.6
excludes a linearly isomorphic copy of ordinary c_0 in Z.
Propositions 4.2 and 4.3 give a bi-Lipschitz bijection
g: E (+) U -> U
with E = c_0(N) and lower constant
c_g = 1/5 - 24η > 0.
Lemmas 6.3 and 6.4 give a bounded linear map Q: Z -> U and a
Lipschitz map K: E (+) U -> Z with
Q K = g - P,
Lip(K) <= 3.
The assumptions of Lemma 2.1 (the absorption criterion) are now
all in place. Applying it yields the onto bi-Lipschitz map
F: Z (+) E -> Z,
F(b, t) = b + K(t, Qb),
with the explicit inverse
G(d) = ( d - K(t_d, u_d), t_d ), (t_d, u_d) = g^{-1}(Qd).
Since E = c_0, this is exactly the map F: Z (+) c_0 -> Z in the
statement of Theorem 1.1. Its inverse is the bi-Lipschitz
bijection in the opposite direction Z -> Z (+) c_0.∎
COROLLARY 8.1 (Same-space consequences). For this same
separable real Banach space Z:
(1) Z (+) c_0 and Z are bi-Lipschitz equivalent but not
linearly isomorphic.
(2) c_0 embeds bi-Lipschitzly into Z, although Z contains no
linear copy of c_0.
(3) Every separable metric space embeds bi-Lipschitzly into
Z.
PROOF of Corollary 8.1.
(1) The direct summand {0} (+) c_0 is a linear isometric copy of
c_0. A bounded linear isomorphism from the direct sum onto Z
would carry it to a closed linear copy in Z, contradicting
Proposition 7.6. The map F from Theorem 1.1 gives the
bi-Lipschitz equivalence.
(2) The restriction t |-> F(0, t) is a bi-Lipschitz embedding.
Aharoni's theorem (cited below, [Aharoni74]) embeds every
separable metric space bi-Lipschitzly into c_0; composing with
this restriction gives a bi-Lipschitz embedding of c_0 into Z.
Since Z has no linear copy of c_0 by Proposition 7.6, this is a
bi-Lipschitz image of c_0 with no linear structure.
(3) Aharoni's theorem embeds every separable metric space
bi-Lipschitzly into c_0; composition with the restriction in (2)
gives the third statement.∎
====================================================================
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END
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