Notation.
N = positive integers {1, 2, 3, ...}
R = reals
sup, inf = supremum, infimum
a_+ = max(a, 0)
|c| = absolute value
||x|| = norm of x; a subscript names the space
c_0 = real sequences tending to 0, norm sup_n |x_n|
c_0(A) = same with index set A in place of N
l_infty = bounded real sequences, sup norm
l_2 = square-summable sequences, norm (sum |x_n|^2)^{1/2}
X (+) Y = direct sum, norm max(||x||, ||y||)
B_X = closed unit ball of X
B = closed unit ball of D (= B_D)
Lip(f) = least L with ||f(x)-f(y)|| <= L||x-y||
supp_B f = closure in B of { s in B : f(s) != 0 }
B_infty(p,r) = { y : ||y - p||_infty <= r }
dist_infty(p,K) = inf_{y in K} ||y - p||_infty
pi_j = D -> R^j, first j coordinates
zeta = B -> Z, zeta(s) = [delta_s - delta_0]
delta_s = evaluation molecule at s (defined below)
Banach spaces are real; 1/2, 2^{-j} mean what they look like.
Brackets [ ] serve only as grouping, interval, or quotient.
A norm ||.|| on a real vector space makes it a metric space with
distance ||x - y||, satisfying
||x|| = 0 iff x = 0, ||c x|| = |c| ||x||,
||x + y|| <= ||x|| + ||y||.
A Banach space is a complete normed space: every Cauchy sequence
has a limit inside, i.e. the space has no holes.
A map T: X -> Y is linear if T(a x + b y) = a T(x) + b T(y).
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 bounded. "X contains a linear copy of c_0" means a closed
linear subspace of X is linearly isomorphic to c_0. That is a
strong property: {0} (+) c_0 sits inside Z (+) c_0 as a linear
copy of c_0, so a linear isomorphism of Z (+) c_0 with Z would
give a linear copy of c_0 inside Z.
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 < +infty. The smallest such L is Lip(F), the
Lipschitz constant of F.
F is bi-Lipschitz if there are 0 < c <= C < +infty 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.
A bi-Lipschitz map can twist, shear and bend distances by at
most a fixed factor, but it can neither collapse distinct points
nor suddenly push nearby points far apart.
Every bounded linear isomorphism is bi-Lipschitz (upper bound
||T||; lower bound 1/||T^{-1}||). The converse is false. In finite
dimensions, bi-Lipschitz equivalence coincides with having the
same dimension (Auerbach's theorem), so counterexamples need
infinite dimensions.
For c_0 itself the gap has been closed. Godefroy, Kalton and
Lancien (2000) proved that a Banach space bi-Lipschitz equivalent
to c_0 (or to a linear subspace of c_0) is linearly isomorphic to
c_0 (or to a subspace of c_0). These theorems concern the whole
domain of the equivalence. One surgical question stays open:
if Y contains a bi-Lipschitz copy of c_0 and Y is not c_0, must Y
contain a linear copy of c_0? Kalton (2008, Problem 2) recorded
this; it is answered negatively below.
pdf/godefroy-kalton-lancien-2000-gafa-c0-lipschitz.pdf
pdf/kalton-2008-rema-nonlinear-geometry-banach.pdf
For Banach spaces X and Y, the direct sum X (+) Y carries the
sup norm ||(x,y)|| = max(||x||, ||y||).
THEOREM 1.1. 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. (a) Z (+) c_0 and Z are bi-Lipschitz equivalent but
not linearly isomorphic (the direct sum holds a linear c_0; Z does
not). (b) Restricting F to {0} (+) c_0 gives a bi-Lipschitz copy of
c_0 inside Z, so Kalton's Problem 2 has a negative answer. (c) With
Aharoni (1974), Z is bi-Lipschitz-universal for separable metric
spaces.
pdf/aharoni-1974-israel-j-math-lipschitz-equivalent-c0.pdf
Infinite-dimensional Banach spaces can be bi-Lipschitz equivalent
without being linearly isomorphic. Aharoni and Lindenstrauss
(1978, non-separable), Ribe (1984, separable), Aharoni and
Lindenstrauss (1985, uniformly convex), and Johnson, Lindenstrauss
and Schechtman (1996) built the stock of examples. The
Godefroy-Kalton-Lancien theorems (2000) showed bi-Lipschitz
equivalence is not far from linear isomorphism when the whole
space is c_0 or a subspace.
pdf/aharoni-lindenstrauss-1978-bams-uniform-equivalence.pdf
pdf/ribe-1984-israel-j-math-separable-uniformly-homeomorphic.pdf
pdf/aharoni-lindenstrauss-1985-israel-j-math-extension-of-ribe.pdf
pdf/johnson-lindenstrauss-schechtman-arxiv-math-9701203-1996.pdf
Start with an abstract algebraic observation.
Suppose E, U, Z are Banach spaces and three maps are given:
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
three constructions, an onto shift g, a test-based space Z, and
the two maps Q and K. Q reads the shift correction from a molecule
and K is a radial Lipschitz lift of s -> [delta_s - delta_0].
A map F: X -> Y between Banach spaces is a bi-Lipschitz embedding
if there are constants
0 < c <= C < +infty, 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.
Ribe (1976) proved that uniformly homeomorphic normed spaces share
the same finite-dimensional linear structure up to one distortion
bound; uniform homeomorphism is weaker than bi-Lipschitz
equivalence. Aharoni and Lindenstrauss (1978) gave nonseparable
bi-Lipschitz equivalent nonisomorphic spaces and asked for
separable examples; Ribe (1984) supplied them.
pdf/ribe-1976-arkiv-uniformly-homeomorphic-normed-spaces.pdf
Johnson, Lindenstrauss and Schechtman (1996) developed both rigidity
results and separable examples with distinct linear structure under
the weaker relation of uniform homeomorphism.
Collectively these results show that a separable counterexample
needs more than a uniform deformation.
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. Its closed subspaces, quotients, and basic
Banach-space properties are completely classified.
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 Theorems 2.1-2.2.
These theorems concern the whole domain of the equivalence. One
question stays open, as stated earlier. Kalton
(2008, pp. 21-22) recorded it as Problem 2, distinct from the
separable Lipschitz-isomorphism question (Problem 3). Hajek,
Johanis and Schlumprecht (2024, arXiv:2401.00831v1; 2025) study
non-linear embeddings into c_0(Gamma). The answer is negative, and
by a stronger route: a separable real Banach space Z exists, with no
linear c_0, that is bi-Lipschitz equivalent to Z (+) c_0 itself.
pdf/hajek-johanis-schlumprecht-2401.00831v1-point-character.pdf
Three earlier constructions control less.
Kalton (2004) built a separable Schur space S (weakly convergent
sequences converge in norm) uniformly homeomorphic to a space with
a complemented c_0 (Theorem 4.6, Propositions 5.1-5.2); its Schur
property excludes a linear c_0, and interleaving the two c_0
summands gives uniform c_0 absorption with nonlinear modulus.
pdf/kalton-2004-collect-spaces-lipschitz-holder.pdf
Sari (2026) embedded c_0 coarse-Lipschitzly into a separable dual
space (Introduction and Theorem 5), allowing additive error;
Theorem 1.1 gives global two-sided Lipschitz bounds and an onto
map, which is strictly stronger.
pdf/sari-2608.04117v2-coarse-lipschitz-c0.pdf
The proof splits into an algebraic reduction and three concrete
constructions. The reduction is elementary and coordinate-free,
specifying exactly what the construction must supply.
The algebraic reduction. Suppose E, U are Banach spaces, and P is
projection from E (+) U onto U:
P: E (+) U -> U, P(t, u) = u.
The construction needs an onto bi-Lipschitz map g: E (+) U -> U;
a bounded linear map Q: Z -> U; and a Lipschitz map K: E (+) U -> Z;
all subject to the exact identity
Q K = g - P. (*)
Any such (g, Q, K) gives 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
(Corollary 3.2 of Godefroy and Kalton 2003), so insisting on
surjectivity of Q would destroy the possibility of a genuinely
nonlinear construction. Requiring only (*) keeps the map Q
potentially non-surjective.
pdf/godefroy-kalton-2003-studia-lipschitz-free-banach-spaces.pdf
The first task is to build g. Take E = c_0(N) and U a countable-
coordinate copy of c_0 built out of finite blocks. The construction
below builds finite-coordinate weights v^lam(s), w^lam(s) and their
localized correction functions f_gam. The blocks of U are arranged
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.
Later, the change of the weights is controlled to get global
bi-Lipschitz bounds, and Brouwer's fixed-point theorem on finite
rows proves 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.
The second task is to 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}. Each depends on
finitely many coordinates and has localized support. Two local
operations enlarge them:
* 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.
The tests are then paired with mass-zero finite combinations
of points ("molecules"), following Arens and Eells (1956,
Section 2) and the Lipschitz-free framework of Godefroy and
Kalton (2003, 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.
The completion and duality properties needed for this norm follow
directly. The coordinate tests define Q, and radial extension of
the point molecules defines K so that (*) holds.
pdf/arens-eells-1956-pjm-embedding-uniform.pdf
The third task excludes a linear c_0 in Z. A hypothetical c_0
basis gives vectors with uniformly bounded signed sums. One
detecting test per vector is extracted from a single class with a
common operation budget; each label records its starting correction
coordinate. Distinct labels separate by tails or by nested
coordinate cylinders. A repeated label leaves, after expansion,
either disjoint tails or a first small local change; the latter
shrinks the support and makes the value small enough that nested
tests still have uniformly Lipschitz signed sums. The norm then
forces the signed sums to grow, a contradiction.
Assembly combines the obstruction with (*) and the
onto shift to prove Theorem 1.1 and its three same-space
consequences.
The passage from a lifted correction to an absorption map is
elementary and coordinate-free; stated first, it specifies what
the construction must supply.
Triangular changes of variables also appear in the quotient-
section construction of Aharoni and Lindenstrauss (1978,
pages 281-282) and in the free-space splitting of Godefroy and
Kalton (2003, Theorem 2.12). The following argument lifts a
correction term and supplies its own inverse formula.
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.
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.
The identity QK(t,u) = g(t,u) - u for the pair (t,u) was used. 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.
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.
The remaining sections supply this criterion with E = c_0. The
inverse above uses only QK = g - P: it never asks for a b with
prescribed Qb until b = d - K(g^{-1}(Qd)) has produced one. That
is what frees the construction from needing Q surjective or
Lipschitz right-invertible, which by Corollary 3.2 of Godefroy
and Kalton (2003) would force a bounded linear right inverse.
The shift will replace one scalar in each coordinate block by a
scalar from the preceding block. First describe the
operation, then choose weights whose dependence on the input is
slow and whose correction coordinates have the support properties
needed for the later tests.
For a countably infinite set A, let c_0(A) have its supremum
norm. Write
N = {1, 2, 3, ...},
index levels by lam = (i, n) in N^2, and put
J_lam = i + n.
For j >= 1 set
d_j = 2^{-j}, A_j = ( [-1, 1] cap d_j Z )^j.
For j = 0, let A_0 consist of one point (the empty tuple).
The finite coordinate block at level lam is
Gam_lam = { (lam, j, xi) : 0 <= j <= J_lam, xi in A_j }.
The three spaces used below are:
E = c_0(N),
U = c_0( disjoint union over lam in N^2 of Gam_lam ),
D = E (+) U.
Write x = (t, (x^lam)_lam) for an element of D, with t in E and
x^lam in R^{Gam_lam} for each lam. Put P(t, u) = u.
Fix an enumeration of the disjoint union of the coordinate sets
of E and U. The map
pi_j: D -> R^j
retains the first j coordinates in this enumeration; pi_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 lam^- = (i, n-1) and
put
r_{i,1}(x) = max( |t_i|, ||x^{i,1}||_infty ),
r_{i,n}(x) = max( ||x^{i,n-1}||_infty, ||x^{i,n}||_infty ).
Each r_lam is 1-Lipschitz. Every input coordinate occurs in at most
two of these maxima. Therefore, for every fixed x in D,
r_lam(x) -> 0 as lam leaves finite sets. (3.1)
Indeed, infinitely many radii at least eps > 0 would require
infinitely many input coordinates of modulus at least eps, and
c_0-type spaces only have finitely many coordinates of modulus
>= eps above any fixed threshold.
The shift will replace one scalar in each block by an incoming
scalar from the preceding block. A "control direction" is needed inside each block. Fix a vector
v in R^{Gam_lam}
and a row (a linear functional)
w in (R^{Gam_lam})^*
with
w v = 1.
For a block vector z in R^{Gam_lam} 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 b is taken from the preceding block's readout,
with t_i entering the first block.
The shift below 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^lam(s)||_infty <= 1, ||w^lam(s)||_1 <= 2,
w^lam(s) v^lam(s) = 1.
(W2) For s, s' in B, put m = min(r_lam(s), r_lam(s')). Then
m * max( ||v^lam(s) - v^lam(s')||_infty,
||w^lam(s) - w^lam(s')||_1 )
<= eta ||s - s'||_D,
where eta = 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 localization. For j >= 1, a
correction coordinate indexed by (lam, j, xi) is supported where
the first j state coordinates are close to xi. Whenever j < J_lam,
it also requires a positive lower bound on r_lam(s) depending only
on j. The first restriction sharpens as j grows; the second gives a
fixed radial threshold when j is fixed and the level varies. Both
bounds hold for the correction coordinates after the weights are
built.
LEMMA 3.1 (Weights at one level). Let eta = 10^{-3}. There are
continuous maps
v^lam: B -> R^{Gam_lam},
w^lam: B -> (R^{Gam_lam})^*,
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
th_j^{J_lam}( r_lam(s) ) and a vector or row that depends only on
pi_j s. If th_j^{J_lam}(r) > 0 then
r <= rho_j (j >= 1),
r >= l_j (j < J_lam), (3.2)
for positive constants rho_j, l_j defined in the proof.
Fix j >= 1 and s in B. For xi in A_j put
b^j_xi(s) = ( 1 - ||pi_j s - xi||_infty / d_j )_+,
B_j(s) = sum_{xi in A_j} b^j_xi(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_xi(s) = b^j_xi(s) / B_j(s),
v^j_xi(s) = min( 1, ( 2 - ||pi_j s - xi||_infty / d_j )_+ ).
The entries are nonnegative. Moreover
||w^j(s)||_1 = 1, ||v^j(s)||_infty <= 1.
If w^j_xi(s) > 0, then ||pi_j s - xi||_infty < d_j and hence
v^j_xi(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_xi and v^j_xi 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.
The proof of Lemma 3.1 continues (theta schedule).
Choose H > 0 with pi/H <= eta/2, and inductively choose numbers
rho_j > 0 so that
0 < rho_1 < 1,
rho_{j+1} < e^{-H} rho_j,
L_j rho_j <= eta/4 (j >= 1).
Each constraint at stage j has a positive right-hand side, so
this choice is possible. Put
l_j = e^{-H} rho_{j+1} (j >= 0).
For an integer J >= 1 define
th^J(r) = ( th^J_0(r), th^J_1(r), ..., th^J_J(r) )
on [0, 1] as follows.
* For r >= rho_1 set th^J(r) = e_0 (the first standard basis
vector of R^{J+1}).
* As r decreases across the interval [e^{-H} rho_j, rho_j], for
1 <= j <= J, rotate from e_{j-1} to e_j by setting
th^J_{j-1}(r) = cos( alp_j(r) ),
th^J_j(r) = sin( alp_j(r) ),
alp_j(r) = (pi / (2H)) * log( rho_j / r ),
with all other entries zero.
* Between these transition intervals, keep the most recently
reached unit vector.
* For 0 <= r <= e^{-H} rho_J set th^J(r) = e_J.
The strict inequality rho_{j+1} < e^{-H} rho_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} th^J_j(r)^2 = 1, (3.3a)
sum_{j=0}^{J} th^J_j(r) <= 2. (3.3b)
For 0 < r <= 1 the path u -> th^J(e^u) is piecewise continuously
differentiable and has l_1 speed at most
(pi / (2H)) * ( |sin alp| + |cos alp| ) <= pi/H.
It is therefore pi/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 th^J_J(0) = 1 is allowed.
For J = J_lam and r = r_lam(s), now define the concatenated weights
v^lam(s) = ( th^J_j(r) v^j(s) )_{j=0}^{J},
w^lam(s) = ( th^J_j(r) w^j(s) )_{j=0}^{J}.
Every band depends on finitely many coordinates: pi_j s and
r_lam(s) involve finite coordinate sets. Continuity includes
r = 0 and transition endpoints. The bounds on the band weights
and (3.3) give
||v^lam(s)||_infty <= 1,
||w^lam(s)||_1 = sum_{j=0}^{J} th^J_j(r) <= 2,
w^lam(s) v^lam(s) = sum_{j=0}^{J} th^J_j(r)^2 = 1.
This proves (W1).
For (W2), put
r = r_lam(s), r' = r_lam(s'),
m = min(r, r'), del = ||s - s'||_D.
If m = 0 there is nothing to prove. Otherwise
m |log r - log r'| <= |r - r'| <= del.
Change the band coefficients first, using the band weights at
s'. The logarithmic Lipschitz estimate and the bounds
||v^j(s')||_infty <= 1 and ||w^j(s')||_1 <= 1 show that this part of
either norm difference, multiplied by m, is at most
(pi/H) del <= (eta/2) del.
Then change the band weights with coefficients frozen at
th^J(r). Only positive j vary, and th^J_j(r) > 0 implies
m <= r <= rho_j. Hence this part of either norm difference,
multiplied by m, is at most
del * sum_{j=1}^{J} th^J_j(r) * m L_j
<= (eta del / 4) * sum_{j=1}^{J} th^J_j(r)
<= (eta del / 2).
The sum of the two contributions proves (W2). This completes
the proof of Lemma 3.1. QED
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 lam = (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 gam = (lam, j, xi), where lam = (i, n) and 0 <= j <= J_lam and
xi in A_j, define the "correction coordinate"
f_gam(s) = th_j^{J_lam}( r_lam(s) ) * v^j_xi(s)
* ( a_{i,n-1}(s) - a_{i,n}(s) ). (3.4)
These f_gam will be exactly the coordinate functions of the
frozen shift correction (S_s - P)s. Preview: the
frozen shift is (S_s x)^lam = x^lam + v^lam(a_{lam^-} - a_lam), so
(S_s - P) x has lam-th coordinate v^lam(a_{lam^-} - a_lam). Distributing
v^lam into its bands via v^lam = ( th_j^J v^j )_{j=0}^J, and
evaluating at x = s, gives exactly f_gam(s) in the gam-coordinate
of U. So each f_gam is one coordinate of the correction.
The next lemma collects everything needed about f_gam.
LEMMA 3.2 (Finite dependence and local support). For every
gam = (lam, j, xi), the function f_gam is continuous, vanishes at 0,
and depends on a nonempty finite set I_gam of coordinates of D.
The sets I_gam can be chosen to include the first j coordinates
of D and every coordinate used by r_lam. Uniformly in gam,
|f_gam(s)| <= 4 r_lam(s) <= 4, (3.5a)
Lip(f_gam) <= 4 + 12eta. (3.5b)
Moreover,
f_gam(s) != 0 implies ( ||pi_j s - xi||_infty <= 2 d_j
if j >= 1,
r_lam(s) >= l_j
if j < J_lam ), (3.6)
where both conditions are imposed when both apply.
PROOF of Lemma 3.2.
For n > 1, a_{i,n-1} uses only
s^{i,n-1}, r_{i,n-1}(s), pi_{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), pi_{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_gam for
all coordinates gam at level lam = (i,n) is the union of:
* the first J_lam enumerated coordinates of D,
* the coordinate t_i,
* every coordinate of the blocks Gam_{i,k} with
k in { n, n-1, n-2 } cap N.
Call this union I_gam. It is nonempty and contains every coordinate
required by the statement.
BOUND |f_gam| <= 4 r_lam(s).
The row bound (W1) gives
| a_{i,n-1}(s) - a_{i,n}(s) | <= 4 r_lam(s),
also when n = 1. Hence
|f_gam(s)| <= 4 r_lam(s) <= 4, and f_gam(0) = 0.
Fix s, s' in B and write del = ||s - s'||_D. At each level lam choose
the frozen pair (v^lam, w^lam) from whichever of s, s' has the larger
local radius r_lam, breaking ties arbitrarily. If e in {s, s'} is
the other endpoint, (W2) gives
r_lam(e) * max{ ||v^lam(e) - v^lam||_infty, ||w^lam(e) - w^lam||_1 } <= eta del.
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^lam(e)( a_{i,n-1}(e) - a_{i,n}(e) ) costs at
most 4etadel from the vector and at most 2etadel 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_lam(s - s') <= 4 del.
Comparing the two endpoints therefore gives
|f_gam(s) - f_gam(s')| <= (4 + 12eta) del.
Suppose f_gam(s) != 0. In the product (3.4) both factors
th_j^{J_lam}(r_lam(s)) and v^j_xi(s) are positive. For j >= 1, the
formula for v^j_xi gives
||pi_j s - xi||_infty < 2 d_j,
which implies the first closed inequality in (3.6). If j < J_lam,
the second inequality follows from (3.2). This covers j = 0 as
well: there the grid condition is dropped, but the lower bound
r_lam(s) >= l_0 still holds. This completes the proof of Lemma
3.2. QED
REMARKS ON THE CONSTANTS. From now on set
A := sup_gam ||f_gam||_infty <= 4,
L_0 := sup_gam Lip(f_gam) <= 4 + 12eta = 4.012. (3.7)
These bounds are uniform in gam. Because eta = 10^{-3}, the
constant 4 + 12eta is comfortably less than 4 + 1/80.
The lemma also pins down exactly where each f_gam 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_lam). This locality is what the constructions in
the argument below will exploit.
The weights of Lemma 3.1 now provide the map required by the
absorption criterion. First invert arbitrary frozen weights,
then bound their variation, and finally solve every target by a
finite-dimensional fixed-point argument.
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||_infty <= 2||y||_infty,
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}||_infty <= ||x^{i,n}||_infty + 4 r_{i,n}(x)
<= 5 r_{i,n}(x).
By (3.1), the block norms r_{i,n}(x) tend to zero as lam goes to
infinity, so Sx is in U and ||S|| <= 5.
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}||_infty, the scalar family (b_{i,n})
belongs to c_0(N^2). Formula (4.2) then gives t_i in 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. QED
For s in B, let S_s be the frozen shift of Lemma 4.1 using
v^lam(s), w^lam(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.
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_gam from Lemma 3.2.
For s in B, define
h(s) = ( S_s - P ) s. (4.4)
Lemma 4.1 guarantees h(s) in U. By (3.4) the gam-coordinate of h(s)
is
( h(s) )_gam = f_gam(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
gam = (lam, j, xi) and every x != 0:
( ( g - P )(x) )_gam
= f_gam( 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 gam-coordinate and invoking (4.4) gives (4.5). QED
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.
PROPOSITION 4.2 (Global distance bounds). For every x, x' in
D,
( 1/5 - 24eta ) ||x - x'||_D
<= ||g(x) - g(x')||_U
<= ( 5 + 24eta ) ||x - x'||_D. (4.6)
In particular, g is injective and continuous.
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 lam, freeze the entire pair
(v^lam, w^lam) at whichever of s_x, s_{x'} has the larger r_lam,
breaking a tie arbitrarily. These choices may vary with lam, but
Lemma 4.1 still applies to the resulting frozen shift Sbar:
Sbar is the frozen shift with pairs (vbar^lam, wbar^lam) at level lam. For
e in { s_x, s_{x'} }, the chosen pair vbar^lam either agrees with the
weight at s_e, or r_lam(s_e) is the smaller of the two radii.
Thus (W2) and (4.7) give
r_lam(s_e) max{ ||v^lam(s_e) - vbar^lam||_infty, ||w^lam(s_e) - wbar^lam||_1 }
<= 2 eta d. (4.8)
This remains valid when r_lam(s_e) = 0 (both sides are then zero
because vbar^lam 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 lam = (i, n). Compare the block lam of g(e) = S_{s_e}(e)
with the block lam of Sbar(e). First change its vector v^lam (from
v^lam(s_e) to the frozen vbar^lam) while keeping the two scalars
computed with the prior rows a_{i,n-1}(e) and a_{i,n}(e). Their
difference has modulus at most 4 r_lam(e), so (4.8) bounds this
change by
8 eta d.
Changing the current row w^lam changes only its scalar readout
a_{i,n}(e), by at most 2etad, because ||e^lam||_infty <= r_lam(e). If n > 1,
changing the predecessor row w^{i,n-1} costs another 2etad, using
(4.8) at (i, n-1) and ||e^{i,n-1}||_infty <= r_{i,n-1}(e). For n = 1,
the predecessor t_i does not change. Since ||vbar^lam||_infty <= 1,
|| ( S_{s_e} e - Sbar e )^lam ||_infty <= 12 eta d. (4.9)
Taking the supremum over lam and adding the two endpoint errors,
|| g(x) - g(x') - Sbar(x - x') ||_U <= 24 eta d. (4.10)
The two frozen bounds of Lemma 4.1 give
( 1/5 ) d <= || Sbar(x - x') ||_U <= 5 d. (4.11)
Combining (4.10) and (4.11),
||g(x) - g(x')||_U <= 5d + 24etad = (5 + 24eta)d,
||g(x) - g(x')||_U >= (1/5)d - 24etad.
The second inequality is (1/5 - 24eta) d, because the frozen
lower bound gives ||Sbar(x-x')||_U >= d/5, and g(x)-g(x') differs
from Sbar(x-x') by at most 24etad. 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 eta, because no
endpoint comparison is needed. Finally, 1/5 - 24eta > 0 because
eta = 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. QED
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, y's block scalars can be read off and fed
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} |Gam_{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.
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 gam-coordinate is
exactly f_gam(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 below 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_gam || f_gam ||_infty <= 4, (4.21)
L_0 := sup_gam Lip( f_gam ) <= 4 + 12eta.
These constants will be used repeatedly below.
What is left: 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
Qzet(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.
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_gam of (3.4) are too simple to build Z from;
each depends only on the coordinates of one block and its
predecessor. The family below enlarges this "tests" by two
local operations and a tail factor, while retaining quantitative
control over their supports and Lipschitz constants.
Fix a label gam = (lam, j, xi) and its nonempty finite set I = I_gam 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 eps = 2^{-a}, a in N_0 = {0, 1, 2, ...}, let
Q_{I,eps} = ( [-1, 1] cap (eps/8) Z )^I.
This is a finite grid that includes both endpoints in each
coordinate. Every point of the cube is within eps/16, hence within
eps/8, of a grid point in the supremum norm.
Define the 2-Lipschitz function
chi(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,eps}, put
(G_{q,eps} u)(s) = u(q) + chi( ||s|_I - q||_infty / eps )( u(s) - u(q) ),
R_{q,eps} u = G_{q,eps} u - u. (5.1)
Both operations are linear in u, retain dependence on I, and
G_{q,eps} u is the constant u(q) on the open cylinder
||s|_I - q||_infty < eps/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,eps} u(q) = u(q), because chi(0) = 0, so
R_{q,eps} u(q) = 0.
LEMMA 5.1 (Bounds and geometric support). For every such u,
q, eps:
||R_{q,eps} u||_infty <= eps Lip(u), (5.2a)
Lip(R_{q,eps} u) <= 3 Lip(u), (5.2b)
Lip(G_{q,eps} u) <= 4 Lip(u), (5.2c)
||G_{q,eps} u||_infty <= ||u||_infty, (5.2d)
||R_{q,eps} u||_infty <= 2 ||u||_infty. (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,eps} and R = R_{q,eps}. With
K_eps = { p : dist_infty(p, K_I(u)) <= eps } and the convention that
empty_eps = empty:
K_I(G u) , K_I(R u) sub K_I(u)_eps, (5.2f)
K_I(R u) sub closure B_infty( q, eps ). (5.2g)
PROOF of Lemma 5.1.
Set
a(s) = 1 - chi( ||s|_I - q||_infty / eps ).
Then
R_{q,eps} u(s) = a(s)( u(q) - u(s) ),
0 <= a <= 1,
Lip(a) <= 2/eps.
If a(s) != 0, then ||s|_I - q||_infty < eps. The cylindrical Lipschitz
bound gives |u(q) - u(s)| <= eps 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,eps} u)(s) = (1 - a(s)) u(s) + a(s) u(q),
so |G(s)| <= max( |u(s)|, |u(q)| ) <= ||u||_infty. This is (5.2d).
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/eps) ||s - t||_D eps 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).
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 <= eps.
* 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||_infty < eps gives
dist_infty(s|_I, K_I(u)) < eps.
Either way, the projection s|_I lies in K_I(u)_eps. Taking closures
gives both inclusions (5.2f).
For (5.2g): Ru(s) = 0 whenever ||s|_I - q||_infty >= eps. So the
nonzero set of Ru is contained in the ball closure B_infty(q, eps) in
the I-cube. Taking closures gives (5.2g).
If u = 0 the statements are immediate. QED
For h >= 1, let T_h consist of functions obtained from a single
f_gam, gam = (lam, j, xi), by m <= h successive operations G or R on
I_gam. A representation with m = 0 is allowed. For m >= 1, the
scales in their order of application must satisfy
eps_1 >= eps_2 >= ... >= eps_m > 0,
sum_{k=1}^{m} eps_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.
Call gam 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 no replacement is made of 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 gam = (lam, j, xi) with
total scale
e = sum_{k=1}^{m} eps_k.
On supp_B u,
if j >= 1: ||pi_j s - xi||_infty <= 2 d_j + e,
if j < J_lam: r_lam(s) >= l_j - e. (5.4a)
In particular, on the same support,
if j >= 1: ||pi_j s - xi||_infty <= 2 d_j + l_j / 2,
if j < J_lam: r_lam(s) >= l_j / 2. (5.4b)
PROOF of Lemma 5.2.
Start with (3.6), which holds on the relative support of f_gam 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 eps. 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_gam} - q||_infty < eps. Otherwise the old support already
contains s.
Both pi_j and r_lam depend only on I_gam. Coordinate restriction has
norm one. And r_lam 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
eps, and the second lower bound decreases by at most eps. 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). QED
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
lam = (i, n):
* the first j coordinates are confined to a cylinder of radius
2 d_j + l_j / 2 around xi (when j >= 1), and
* the local radius r_lam is bounded below by l_j / 2 (when
j < J_lam).
The two alternatives - small j (bounded bands) versus large j
(unbounded bands) - will dominate the case analysis below.
Bounded bands give a positive local radius threshold. Unbounded
bands concentrate the first j coordinates to a small cylinder.
Suppose a test in T_h has support concentrated where the local
radius r_lam 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 localize independently in either
direction, without disturbing the band structure.
For N >= 1, define the coordinate tail
tau_N(s) = sup_{k > N} |s_k|,
b_{c,N}(s) = min( 1, ( 2 - tau_N(s)/c )_+ ) (c > 0). (5.5)
Write a_+ = max(a, 0). The function tau_N is 1-Lipschitz in s (with
respect to the supremum norm on D), and
tau_N(s) -> 0 for each s in B as N -> infty.
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} sub { tau_N <= 2c }, (5.6a)
supp_B (b_{c,N} - 1) sub { tau_N >= c }. (5.6b)
VERIFICATION. If tau_N(s) >= 2c, then 2 - tau_N(s)/c <= 0, so the
positive part is 0 and b_{c,N}(s) = 0. That gives (5.6a). If
tau_N(s) <= c, then 2 - tau_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:
tau_N is 1-Lipschitz, tau |-> (2 - tau/c)_+ is 1/c-Lipschitz, and
min(1, .) is 1-Lipschitz.
Let V_h consist of all products
u prod_{r=1}^{l} bet_r (5.7)
where
* u in T_h has a representing label with j <= h,
* 0 <= l <= h,
* each bet_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). 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 below. 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||_infty 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:
|prod_r bet_r(s) - prod_r bet_r(t)|
<= Sig_r (prod_{r' < r} |bet_{r'}(t)|) |bet_r(s) - bet_r(t)|
(prod_{r' > r} |bet_{r'}(s)|)
<= Sig_r Lip(bet_r) ||s - t||_D
<= l 2^h ||s - t||_D.
The product rule then gives
Lip( u prod_r bet_r )
<= Lip(u) sup(prod_r bet_r) + sup(u) Lip(prod_r bet_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. QED
Note the key structural features the argument below uses.
* For a fixed bounded set of bands j, the second line of (5.4b)
gives a positive radial threshold r_lam(s) >= l_j / 2 on the
support of any test with band j (and j < J_lam). Because l_j
depends only on j and not on the level lam, this is uniform in lam
for fixed j.
* For j -> infty, 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.
The argument below uses precisely these alternatives - bounded versus
unbounded bands - to separate or nest the supports of a family of
"detecting" tests.
This completes the test construction. The Banach space Z is built
from these tests and shown to have no linear copy of c_0.
Now turn to the selected tests as 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.
Let M_0(B) be the real vector space of mass-zero finite formal
sums
m = sum_{s in B} a_s del_s,
#{ s : a_s != 0 } < infty, 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. Arens and Eells (1956, Section 2) put it
as: a molecule is "a real-valued function m on X which is zero
except (perhaps) at a finite number of points x_1, ..., x_k of X
and which satisfies sum m(x_i) = 0". This follows the mass-zero
molecule construction of
Arens and Eells (1956) and its later Lipschitz-free formulation
by Godefroy and Kalton (2003). 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
sig_d(m) = sup {
( sum_{nu=1}^{l} | <u_nu, m> |^p )^{1/p} :
l >= 0,
u_1, ..., u_l in L,
Lip( sum_{nu=1}^{l} e_nu u_nu ) <= C
for every (e_nu) 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_{nu=1}^{l} | u_nu(s) - u_nu(t) | <= C ||s - t||_D
for all s, t in B. (6.3)
Indeed, if (6.3) holds, choose signs matching the increments
u_nu(s) - u_nu(t); the triangle inequality gives
Lip( sum_{nu=1}^{l} e_nu u_nu ) <= 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
alp_d = 2^{-d} / C_d^2,
||m||_0 = ( sum_{d >= 1} alp_d sig_d(m)^2 )^{1/2}. (6.4)
The outer square sum and the exponents approaching one serve
distinct purposes below. 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
zet : B -> Z, zet(s) = [ del_s - del_0 ] (6.5)
is 1-Lipschitz with zet(0) = 0. Each sig_d descends and extends
continuously to a seminorm on Z, and the exact formula
||z||_Z^2 = sum_{d >= 1} alp_d sig_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_nu along the segment from s to t. Since the l_p norm is no larger
than this sum,
sig_d( del_s - del_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 del_s of mass zero equals
sum_s a_s ( del_s - del_0 ),
because summing the mass-zero gives zero. Therefore
sig_d(m) <= C_d sum_s |a_s| ||s||_D.
This proves finiteness of sig_d, and hence of ||m||_0, because
sum_{d >= 1} alp_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.
sig_d IS 1-LIPSCHITZ ON zet(B). Equation (6.7) and the same square sum
gives
||del_s - del_t||_0 <= ||s - t||_D.
This proves zet is 1-Lipschitz in (6.5) and zet(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 zet makes zet(B_0) dense in zet(B). The real
linear span of zet(B) is the quotient molecule space; its rational
span over B_0 is consequently dense in Z. Hence Z is separable.
sig_d DESCENDS AND EXTENDS. For molecules m, m':
|sig_d(m) - sig_d(m')| <= sig_d(m - m')
<= alp_d^{-1/2} ||m - m'||_0. (6.8)
In particular sig_d vanishes on N, descends, and extends
continuously to Z. Subadditivity and homogeneity pass to limits.
EXACT FORMULA (6.6). The map
m |-> ( sqrtalp_d sig_d(m) )_{d >= 1}
from M_0(B) into l_2 is 1-Lipschitz, because the coordinate
differences are bounded by sqrtalp_d sig_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 sqrtalp_d sig_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. QED
The obstruction argument below 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> | <= sig_{d_L}(m) <= alp_{d_L}^{-1/2} ||m||_0. (6.10)
The pairing vanishes on N (because N has zero molecule norm and
sig_{d_L} does) and extends to Z^*. Let
B_L = alp_{d_L}^{-1/2}.
Then ||u||_{Z^*} <= B_L for every u in L, uniformly.
EXTENSION OF sig_d TO Z. For an admissible list
u = ( u_1, ..., u_l )
of a fixed triple d, its evaluation map
Lam_u m = ( <u_nu, m> )_{nu=1}^l in l_p^l
has operator norm at most alp_d^{-1/2} on the molecule quotient, by
the same argument as (6.10) applied to the list formula (6.2).
Therefore Lam_u extends with that bound to Z.
The supremum of || Lam_u z ||_p over all admissible lists is finite
and alp_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 sig_d on the dense molecule quotient. It is therefore
the continuous extension of sig_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 -> infty.
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. QED
LEMMA 6.3 (The c_0-valued correction map). There is a bounded
linear map Q: Z -> U such that
Q zet(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_gam 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_gam) is admissible (since f_gam in T_1 and
Lip(f_gam) <= L_0 <= C_0), whence
|| Q_0 m ||_U = sup_gam | < f_gam, m > |
<= sig_{d_0}(m)
<= alp_{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
del_s - del_0 gives
Q zet(s) = Q( del_s - del_0 ) = Q_0( del_s - del_0 )
= h(s) - h(0) = h(s).
This proves (6.11). QED
LEMMA 6.4 (Lipschitz lift). Define
K(0) = 0,
K(x) = ||x||_D * zet( 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 zet. Since zet(0) = 0 and Lip(zet) <= 1 (Lemma 6.1),
|| zet(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 zet(s):
|| K(x) - K(x') ||_Z
= || a zet(s) - b zet(s') ||_Z
<= ( a - b ) || zet(s) ||_Z + b || zet(s) - zet(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 lam > 0 and x != 0,
K(lamx) = ||lamx||_D zet( lamx / ||lamx||_D )
= lam ||x||_D zet( x / ||x||_D )
= lam K(x).
If x = 0, both sides are 0.
THE IDENTITY (6.13). For x != 0,
Q K(x) = ||x||_D Q zet( 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. QED
What is available. 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 Z is known to have no linear copy of c_0. That
is the remaining task.
A linear copy of c_0 in Z would provide vectors bounded away from
zero whose finite signed sums are uniformly bounded. First show
that one class of tests detects a subsequence of these vectors. 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.
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 sub 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 del > 0 with
| < F_i, m_i > | >= del. (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)| < infty, 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 sig_d(m_i) -> 0 for every d. Then the vectors
v_i = ( sqrtalp_d sig_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} alp_d sig_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 sig_d^2 is even and convex, because
sig_d is a seminorm and squaring is convex and increasing on
[0, infty). For fixed y, m,
( sig_d(y+m)^2 + sig_d(y-m)^2 ) / 2 >= sig_d(m)^2. (7.4)
For independent random signs on a finite set H sub 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 sig_d( sum_{k in H} e_k m_k )^2 >= sig_d(m_i)^2,
the expectation over the uniform distribution on signals in
{-1, 1}^H. So for each fixed d,
E sig_d( sum_{k in H} e_k m_k )^2 >= sig_d(m_i)^2
for every i in H.
ASSEMBLING. The inequality (7.1) already holds for every sign
choice, so 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 alp_d E sig_d( sum_{k in H} e_k m_k )^2
>= sum_{i in H} sum_{d in A_i}
alp_d sig_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),
sig_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
sig_{d'}(m_i) <= alp_{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_nu = |<u_nu, m_i>|. The same list is
admissible for d' because only the exponent changed. Therefore
(a/2)^p <= sum_nu x_nu^p
<= ( max_nu x_nu )^{p - p'} sum_nu x_nu^{p'}
<= ( max_nu x_nu )^{p - p'} B_0^{p'}.
The list is nonempty and B_0 > 0. One of its tests therefore has
evaluation at least
del = ( (a/2)^p / B_0^{p'} )^{1/(p - p')} > 0.
Choosing that test as F_i proves (7.2). QED
Call a sequence of tests "compatible" if all its finite signed sums
have a common Lipschitz bound. The norm sig_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 del > 0 and an integer C >= 1 such that
| < F_i, m_i > | >= del 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
>= sqrtalp_d . del . sqrt|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
sig_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
>= del^2.
Summing over k in H,
E sig_d( S )^2
>= sum_{k in H} E | <F_k, S> |^2
>= sum_{k in H} del^2
= |H| del^2.
Combining with the exact norm formula (6.6),
E || S ||_Z^2
= sum_{d' >= 1} alp_{d'} E sig_{d'}( S )^2
>= alp_d E sig_d( S )^2
>= alp_d |H| del^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
>= alp_d |H| del^2.
Taking square roots gives (7.5). QED
Localizing detecting Lipschitz functions and combining them with a
common Lipschitz bound is also central to Kalton's arguments for
metric free spaces (2004, Lemma 4.5 and proof of Theorem 4.6).
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 sub B and a scalar e_i such that u_i = e_i on C_i and
supp_B u_k sub C_i for every k > i. If
sum_i | e_i | Lip(b_i) < infty,
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 sub 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. QED
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> | >= del > 0,
supp_B F_i sub { tau_{M_i} >= c_* }, M_i in N, M_i -> infty,
c_* > 0. (7.7)
Then a subsequence has modified tests in one class V_H, with
detection at least del/2, whose relative supports are pairwise
disjoint.
PROOF of Lemma 7.4.
Choose a dyadic c = 2^{-a} with 2c < c_*. If the prior 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
tau_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 -> infty.
Choose N large enough that the modulus of the difference is at
most del/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 such index so far out that its
M_k exceeds all previously selected N_i (this is possible because
M_k -> infty). 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 tau_{N_i} <= 2c
by (5.6a). The new k-th support requires tau_{M_k} >= c_* by (7.7),
hence
tau_{N_i} >= tau_{M_k} >= c_* > 2c,
because N_i < M_k and tau_N(s) = sup_{j > N} |s_j| is nonincreasing
in N. This contradicts the i-th support condition tau_{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
del - del/2 = del/2. QED
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 > | >= del > 0 (i >= 1).
There are a subsequence (m_{i_k}), a listed class L_1, tests
Ftilde_k in L_1, a constant del_1 > 0, and an integer C_1 >= 1 such
that
| < Ftilde_k, m_{i_k} > | >= del_1,
Lip( sum_{k in H} e_k Ftilde_k ) <= C_1
for every k, every finite H, and all signs e_k.
PROOF of Proposition 7.5.
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
gam_i = ( lam_i, j_i, xi_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. These three possibilities are handled separately.
Every block Gam_lam is finite, so pairwise distinct labels force
lam_i to leave every finite set. Since
{ lam : J_lam <= J }
is finite for every J, J_{lam_i} -> infty. Pass to a further
subsequence with one fixed band j. Eventually j < J_{lam_i}, and
(5.4b) gives
r_{lam_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 -> infty
so that every coordinate used by r_{lam_i} has index > M_i. Then
r_{lam_i}(s) <= tau_{M_i}(s) for every s in B.
The condition r_{lam_i} >= c_* is the hypothesis of Lemma 7.4, so
this gives 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.
Now j_i -> infty. 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
xi_i in [-1, 1]^{j_i}
converge to a single vector
v_* in [-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 ||_infty
<= a_i + || xi_i|_I - v_*|_I ||_infty -> 0. (7.14)
CYLINDER THE TEST. For each fixed i, consider arbitrarily small
positive dyadic th, chosen below the last existing scale if there
is one and with th less than the remaining strict budget in (5.3).
Choose a grid center q_i(th) within th/8 of v_*|_{I_{gam_i}}. The
test
F_{i,th} = G_{q_i(th), th} F_i
belongs to T_{h+1}, as does F_i, and
|| F_{i,th} - F_i ||_infty <= th Lip(F_i) <= th 4^h L_0.
So F_{i,th} -> F_i pointwise as th down 0, even though the centers
move. For the fixed i, Lemma 6.2 and (6.9) allow a choice th_i
with
| < F_{i, th_i} - F_i, m_i > | < del / 2.
This is a separate choice after fixing m_i, not a uniform limit
over the sequence of vectors. Write Ftilde_i = F_{i, th_i}. It detects
m_i with bound del/2.
PLATEAU ON A CYLINDER. The function Ftilde_i is constant on the
relatively open cylinder
C_i = { s in B :
|| s|_{I_{gam_i}} - v_*|_{I_{gam_i}} ||_infty < 3 th_i / 8 } (7.15)
because the distance from its grid center is then less than
th_i / 2. Its enlarged representation still satisfies the strict
budget, so (5.4b) applies to Ftilde_i. Therefore (7.14) holds with
Ftilde_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 = Ftilde_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(Ftilde_i) bound
from Lemma 5.3 applied to class T_{h+1}:
Lip( sum_{k in H} e_k Ftilde_k ) <= 4^{h+1} L_0.
Set C_1 equal to this constant.
Now a single label gam = (lam, j, xi) is repeated. All cylindrical
factors depend on the same finite set I = I_gam. 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 G/R types
of the cylindrical operations, plus the number l <= h and the
types of the tail factors. At each operation position the dyadic
scale has a subsequence either constant or tending to zero; in the
constant case the center can also be made constant, since the
permitted grid in the fixed cube is finite. At each tail position
the scale c can be made constant and its index N is either
constant or tending to infinity. Make these finitely many
extractions together.
EXPANSION AND DETECTION. An ingredient is "fixed" when all its
parameters are constant. All parameters in a concrete
representation are frozen before the algebraic expansion: replace
every nonfixed G_{q,eps} by Id + R_{q,eps} and every nonfixed
factor b_{c,N} by 1 + (b_{c,N} - 1). The operations are linear in
their inputs, so the identities distribute through later operations
with no parameters recomputed. The expansion covers the cylindrical
operators and the tail product separately; G is never applied 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 prior 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
prior 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 ingredients are all fixed is a fixed
concrete test F^a; its pairing with m_i tends to zero by weak
nullity (it need not vanish identically). Finitely many such
positions occur, so their total pairing tends to zero. In view of the detection |<F_i, m_i>| >= del, for all
sufficiently large i, the sum of the absolute pairings of the
varying terms in (7.23) is at least del/2. One of the at most K_0
positions has pairing at least del/(2K_0); a subsequence fixes that
position. Relabel its tests as
F_i = u_i b_i, | < F_i, m_i > | >= del_0 > 0. (7.24)
ESCAPING TAIL INDEX. If b_i contains a nonfixed factor
b_{c, N_i} - 1 with N_i -> infty, its support lies in
{ tau_{N_i} >= c } by (5.6b). The band is fixed, so Lemma 7.4
applies; assume no such factor remains. Every retained nonfixed
tail factor has this form after expansion, so the remaining tail
product is fixed and the variation in (7.24) lies 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
eps_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_gam. 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, eps_i} p ||_infty <= L_pre eps_i,
K_I( R_{q_i, eps_i} p ) sub closure B_infty( q_i, eps_i ).
Every retained later scale is at most eps_i, by the nonincreasing
order in the prior 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 ||_infty <= 2^h L_pre eps_i,
{ s|_I : s in supp_B u_i } sub K_I(u_i)
sub closure B_infty( q_i, (h + 1) eps_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. 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 th that are below the
last retained scale, below the remaining strict budget, and at
most eps_i. Let q_i^+(th) be a permitted grid point within th/8 of
v, and set
F_{i, th} = ( G_{q_i^+(th), th} 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, th} - F_i ||_infty <= th Lip(u_i).
For the fixed vector m_i, pointwise convergence and (6.9) give
an allowed choice th_i such that
| < F_{i, th_i} - F_i, m_i > | < del_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^+(th_i), th_i} u_i and F_i := u_i b_i.
This F_i detects m_i with bound del_0 / 2. The support enlargement
in Lemma 5.1, (7.25), and th_i <= eps_i give
{ s|_I : s in supp_B u_i } sub closure B_infty( q_i, (h + 2) eps_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 eps_i -> 0, these cylindrical supports
concentrate at v.
PLATEAU VALUES. On the relatively open cylinder
C_i = { s in B : || s|_I - v ||_infty < 3 th_i / 8 },
the function u_i is the constant
e_i = u_i( q_i^+(th_i) ).
Equation (7.25) gives
| e_i | <= 2^h L_pre eps_i -> 0. (7.28)
NESTED PLATEAUS. Choose a recursive subsequence on which
sum_i | e_i | < infty,
and
supp_B u_k sub 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) < infty.
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. QED
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 < +infty 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 sub 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} > | >= del > 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
del_1 > 0, and an integer C_1 >= 1 such that
| < F_k, m_{i_k} > | >= del_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, del = del_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
>= sqrtalp_{d_1} . del_1 . sqrt|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 >= sqrtalp_{d_1} . del_1 . sqrt|H| for every finite H.
Since alp_{d_1} > 0 and del_1 > 0, the right-hand side grows without
bound as |H| grows. This is a contradiction. Hence no such
embedding T exists. QED
Collect everything now 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 - 24eta > 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. QED
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 (1974) 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. QED