The cotype–cotype conjecture under the approximation property

Proof. The upper bound follows from the triangle inequality and Cauchy–Schwarz; the lower bound follows by taking input supported at one u, scaled by 2^{n/2}. On V_{n+j}=V_n\oplus V_j, first apply W_n^T in the first coordinate and then the normalized Walsh transform with the identity of F in the second coordinate. The latter has norm at most 1. Averaging the squared estimates in the unused coordinates proves monotonicity. Interchanging finite L_2 coordinates proves (ii), with constant functions on \Omega giving the reverse inequality. For (iii), factor a finite-rank T as G\to H\to F, with H a finite-dimensional Hilbert space. Orthogonality of the Walsh matrix gives \|W_n^{\Id_H}\|=2^{-n/2}, proving (iii). Finally, W_n^A-W_n^B=W_n^{A-B}, so the upper bound in (i) and the reverse triangle inequality prove (2.3).□

Proof. Put M=\E\|S\| and b=\max_i\|z_i\|. Since z_i=\E(\eps_iS), we have b\le M. The case M=0 is immediate.

Let S_j=\sum_{i\le j}\eps_i z_i, and stop at the first j with \|S_j\|>u. Conditional on a stopped prefix z, the remaining sum is symmetric; at least one of \|z+h\| and \|z-h\| exceeds u. Hence

For any fixed j, symmetry of the independent prefix similarly gives

At the first stop, \|S_j\|\le u+b. Decomposing according to its time and using independence of the remaining sum therefore yields

Take u=8M. Markov’s inequality and b\le M imply

Thus \mathbb P(\|S\|>9jM)\le2^{-j} for j\ge0, and integration over consecutive intervals of length 9M gives

□

Proof. The moment map

is onto: a right inverse is (x_i)\mapsto\sum_i\eps_i x_i. Its adjoint sends (z_i^*) to \sum_i\eps_i z_i^*. By Lemma 2.2 and (2.4),

Equation (2.8) bounds the infimal lift norm of (ax_i) by LD_ga\sqrt m=1/2. A lift of norm at most 1 therefore exists, proving (i).

For (ii), the map

satisfies \|Vw\|_{L_1}\ge(LD_f)^{-1}\|w\|_{\ell_2^m(F)}. The functional

has norm at most b\sqrt m\,LD_f=1/2 on its range. Hahn–Banach extends it to L_1(F) with the same norm. Representing the extension in L_\infty(F^*) gives Z and the stated moments. The amplification assertion follows from (2.6) and finite-sum duality.□

Proof. We first reduce the uniform lower bound to a nonvanishing assertion. If every index in the Fourier support of \Psi had rank at most one, a damped polynomial would encode their rectangular supports. Its prescribed mixed derivative will contradict a bound obtained by splitting it into two parts, each of low degree in one variable.

The admissible triples (\Psi,f,g) form a compact set in a finite-dimensional space. The maximum of |\widehat\Psi(P)| over \rank P\ge2 is continuous. It therefore suffices to show that this maximum never vanishes. If the admissible set is empty, any positive c_* will do.

Suppose, to the contrary, that the Fourier support of \Psi has rank at most one. Oddness eliminates indices with an even number of ones. Every remaining index has a unique form P=ph^t with p,h\ne0, and both k=|p| and l=|h| are odd. Let

Parseval’s identity and Cauchy–Schwarz give

To suppress rectangles that are large in both directions, consider the damped polynomial

It has sup norm at most M_0 on [-1,1]^2. To see this, take independent signs D_i,E_j,R_{ij} with means s,u,1/2, respectively, and set S_{ij}=D_iE_jR_{ij}. For an odd rectangle ph^t,

so H(s,u)=\E\Psi(S).

Delete from H the terms with both k>d and l>d, obtaining H_0. If k,l>d, then

Using (3.4), the deleted part is bounded in sup norm by

Consequently \|H_0\|_\infty\le2M_0.

For an odd rectangle ph^t, the vectors of row and column parities are p and h. Comparing Fourier coefficients in (3.3) therefore gives

The deletion defining H_0 does not affect terms with k=1 or l=1. Hence

Independent signs of common mean u/2, or s/2, show that the first two expressions have sup norms at most AM_0 and BM_0.

Every retained monomial has low degree in at least one variable. We separate these two possibilities with controlled sup norms. Expand H_0 in the Chebyshev basis T_j(s), defined by T_j(\cos\theta)=\cos(j\theta). Let H_1 retain the terms with j\le d, and set H_2=H_0-H_1. Then

Indeed, every monomial of H_0 has s-degree at most d or u-degree at most d, and truncation in the Chebyshev basis annihilates every monomial of the former kind in H_2.

Each Chebyshev coefficient of a polynomial on [-1,1] is bounded by twice its sup norm, by cosine orthogonality. Since |T_j|\le1,

Also, |T_j'(0)|\le j, so every polynomial Q of degree at most d satisfies

First apply (3.7) in s to H_1, and in u to H_2. Then (3.5) implies

By (3.6), these are polynomials of degree at most d in their displayed variables. Applying (3.7) once more and adding yields

On the other hand, (3.1) gives

Since 5(d+1)>1+1/(4M_0), these inequalities contradict (3.5) and (3.8).□

Proof. Consider two pairs (U,V), (U',V') with the same pairing matrix. The covector frames define surjections \pi,\pi':V_n\to\F_2^s. The isomorphism u_i\mapsto u_i' between the primal spans respects evaluation, and hence maps \Span(U)\cap\ker\pi onto \Span(U')\cap\ker\pi'. Extend this restriction to an isomorphism between the two kernels. Together with the map on the primal spans, it defines an isomorphism

satisfying \pi'H=\pi. Complete it to an invertible map of V_n by choosing matching lifts of a basis complementary to the common evaluation image \pi(\Span(U))=\pi'(\Span(U')). Then HU=U' and \pi'H=\pi, the latter being equivalent to H^{-t}V=V'. For feasibility in dimension r+s, given a matrix D, take u_i=(e_i,0) and v_j=(D_{:,j},e_j).□

Proof. Put n'=n-2m. We will turn a Walsh test at index n into bounded scalar functions satisfying Lemma 3.1. A coefficient of binary rank k\ge2 will then give a Walsh test at index kn'.

Step 1: realize the Walsh signs exactly and equivariantly. Choose families p_u\in G, q_v\in F^* admissible in (2.1), changing a sign if necessary, such that

Amplify T over V_n\times V_n, and define

Uniform translation shows that every x'_u and y'_v has norm at most one in the corresponding finite L_2 space. Writing T' for the amplified operator, we obtain

Indeed, the pairing includes expectation over \alpha,\beta, and its phase is

Amplify once more over \mathscr G=\GL(n,\F_2), obtaining \widetilde T:\widetilde G\to\widetilde F, and set

For the bilinear form B(x,y)=y(\widetilde T x), this gives

Use the same right translations \Lambda(H)z(R)=z(RH) on the domain and target dual. They are isometries, preserve B under simultaneous translation, and satisfy

The orders follow directly from (RH)^{-t}=R^{-t}H^{-t}. By Lemma 2.1, w_j(\widetilde T)=w_j(T) for every j. Equation (2.6) also preserves the bounds D_g,D_f, so the moment lifts remain available with the same a,b.

Step 2: lift frames while preserving the symmetry. For every primal m-frame U=(u_i)_{i=1}^m, choose P_{U,\eps}\in B_{\widetilde G} with

We can make these choices equivariant. For one reference frame, take the lifts from Lemma 2.3 and average each of them over its stabilizer under \Lambda. The prescribed moments are fixed by this stabilizer, and the averaged lifts are stabilizer-invariant. Transport them to any other frame by \Lambda(H). Two transporters differ on the right by a stabilizer element, so the result is well-defined and equivariant.

The same construction for covector frames, using the action H^{-t}, gives Z_{V,\delta}\in B_{\widetilde F^*} with

The two systems of lifts can be chosen independently. Their equivariance, invariance of B, and Lemma 4.1 show that the following pairings depend only on the displayed sign patterns and lift labels:

All patterns are feasible since n>4m. The moment identities and (4.3) give

Step 3: control the odd parts on a smaller Walsh cube. We need a bound that remains useful after division by \gamma. For every fixed choice of lift labels, we claim that

Decompose V_n=V_{n'}\oplus\F_2^m\oplus\F_2^m, and fix a sign matrix S_0=((-1)^{D_{ij}}). For u,v\in V_{n'}, take the primal frame

and the two covector frames

where \mathbf1\in\F_2^m is the all-ones vector. The tags e_i,e_j ensure independence for every u,v. Their sign matrices are \chi(u,v)S_0 and -\chi(u,v)S_0, so

Both vector families lie in their respective unit balls, and each depends only on its own Walsh index. Testing (2.1) at n', and using amplification invariance, proves the first bound.

For \phi, retain U(u) and use the nonzero covectors (v,h,e_1), (v,h+\mathbf1,e_1), where s_i=(-1)^{h_i}. Half the difference of the corresponding y’s gives the same calculation. For \psi, use the covector frame (v,0,e_j) and half the difference of the primal vectors at (u,e_1,h), (u,e_1,h+\mathbf1). Their pairings are \chi(u,v)(-1)^{h_j} and its negative. The e_1 tags keep the singleton vectors nonzero. This proves (4.6).

Step 4: find a Fourier coefficient of binary rank at least two. Define

These functions are odd, and the slow-decrease hypothesis gives

Changing variables s'=\eps s in the first coefficient and using (4.5), we obtain

The odd projection leaves this coefficient unchanged. Likewise, \widehat g(e_j)=b.

We next check the row and column projections required by Lemma 3.1. A first row projection is unchanged by applying global odd projection, since

Substitute \xi=\eps\sigma; then \xi,\eps,\delta are still independent uniform sign vectors. By (4.5),

Interchanging rows and columns, and simultaneously (a,\eps,\phi) with (b,\delta,\psi), gives \E_\tau\tau_j\Psi(S\tau)=b f(S_{:,j}). Thus all hypotheses of Lemma 3.1 hold, with the fixed parameters m,a,b.

Choose a binary matrix P of rank k\ge2 with |\widehat\Psi(P)|\ge c_*. Since \Psi is odd and this coefficient is nonzero, the total number of ones in P is odd. Its coefficient is therefore unchanged when the odd projection is removed from (4.7). Writing \Phi_{\eps,\delta}(S)=\Phi(S,\eps,\delta), row and column sign multiplication gives the exact identity

For some fixed \eps,\delta, therefore, h=\Phi_{\eps,\delta} satisfies

No oddness assertion about this individual function h is needed.

Step 5: extract the independent Walsh factors. Keep P,\eps,\delta fixed. For arbitrary tuples (u'_i)_{i=1}^m, (v'_j)_{j=1}^m in V_{n'}, set

where D ranges over binary m\times m matrices. The tagged tuples are frames even if the original tuples have repeated or zero entries. The Fourier coefficient formula gives

where \langle P,D\rangle=\sum_{i,j}P_{ij}D_{ij} is computed modulo two. Indeed, translate the uniform matrix D by (u'_i\cdot v'_j)_{ij} in the defining Fourier average. Both vectors in (4.9) have norm at most one. The first depends only on the u'-tuple, and the second only on the v'-tuple.

Since \rank_{\F_2}P=k, choose binary m\times k matrices I,J with I^tPJ=\Id_k. For \zeta,\eta\in(V_{n'})^k, substitute

The exponent in (4.9) becomes \sum_{\ell=1}^k\zeta_\ell\cdot\eta_\ell. We have thus obtained a Walsh test on V_{kn'}, of amplitude \widehat h(P), with separate unit-ball families on its two sides. Monotonicity and amplification invariance now yield

Taking c_1=\min(1/2,c_*/2) proves the lemma.□

Proof. Let c_1 be given by Lemma 4.2. Choose an integer K>\log_2(1/c_1), and put

Fix a nonzero finite-rank T, and suppose that (4.1) holds at some n\ge n_0. The growth lemma gives

By monotonicity, the same lower bound holds throughout [n,2n-4m].

Set a_n=\lceil3n/2\rceil. The K consecutive blocks

fit in this interval, since 2n-a_n=\lfloor n/2\rfloor\ge2mK+4m+1. Their endpoint ratios telescope:

Since 2^K>1/c_1, one right endpoint t satisfies

We can repeat the argument at t. Every application of the growth lemma returns an inequality for the original T; its auxiliary amplifications are discarded. We therefore obtain indices n_j\ge(3/2)^j n with w_{n_j}(T)\ge c_1^j w_n(T)>0. The rank-dependent estimate (2.2), for this one fixed operator, implies

a contradiction. Thus (4.1) is impossible for n\ge n_0, proving the strict form of (4.10) for nonzero finite-rank T. The zero operator satisfies the asserted weak inequality.

Iteration proves (4.11); for example, take

Finally, if finite-rank maps T_j converge in operator norm to T, then (2.3) gives w_s(T_j)\to w_s(T) for every s, while \|T_j\|\to\|T\|. Passing to the limit proves both estimates for T.□

Proof. Finite-dimensional enlargements. We construct nested finite-dimensional subspaces

such that each z=(z_i)_{i=1}^m\in\ell_2^m(E_l) has a lift H_l:\{-1,1\}^m\to E_{l+1} satisfying

First take moments in the whole space X. The map

has the bounded right inverse (z_i)\mapsto\sum_i\eps_i z_i, and

By (2.8), the infimal lift norm of a unit tuple is at most \Delta, so a lift of norm below 2\Delta exists. Take a finite \eta-net on the unit sphere of \ell_2^m(E_l), where 0<\eta\le\min(1/2,\Delta/\sqrt m), and choose such a lift for each net point. Let E_{l+1} be the span of E_l and all the values of these finitely many lifts.

For any unit tuple z\in\ell_2^m(E_l), choose a net point v with \|z-v\|_{\ell_2^m(X)}\le\eta. Add \sum_i\eps_i(z_i-v_i) to its lift. This corrects the moments exactly and adds at most

to the norm. This proves (5.2) for unit tuples; scaling and the zero lift cover the other cases.

The compact map. Put

Cauchy–Schwarz proves absolute convergence of the series and gives

The same estimate on the tails shows that the finite-rank truncations converge in operator norm. Thus j is compact. Let G=\mathcal H/\ker j, with quotient map Q, and define J by JQ=j. The kernel is closed, so G is Banach; the quotient norm gives \|J\|\le\|j\|<2, and J is injective. Every element of B_G has a representative in 2B_{\mathcal H}. Hence J(B_G)\subset j(2B_{\mathcal H}) is relatively compact, which proves compactness of J. Finally, \iota(x)=Q(x,0,0,\ldots) is a contraction and J\iota x=x. The finite-rank truncations of j need not annihilate \ker j, so they need not induce operators on G. Thus this compactness argument does not itself approximate J by finite-rank maps; that passage will use the approximation property of X.

Dual moments from the weighted shift. The nesting of the E_l’s makes

a bounded map. The factor 2 compensates for the next dyadic weight, so jR=j and therefore QR=Q. This identity lets us assemble the finite-level lifts into exact moment lifts in G.

For g_1,\ldots,g_m\in G, choose representatives z^i=(z_l^i)_l\in\mathcal H with Qz^i=g_i and \|z^i\|\le2\|g_i\|, taking z^i=0 when g_i=0. Use (5.2) on each tuple (z_l^i)_i, and set

Summing squared estimates yields

Every atom of the sign cube has positive mass, so this estimate also shows that \widetilde H(\eps)\in\mathcal H for every \eps. Its i-th moment is Rz^i. Since QR=Q,

For \theta_1,\ldots,\theta_m\in G^*, it follows that

Taking the supremum over \sum_i\|g_i\|^2\le1 gives c_m(G^*)\le12\Delta by finite \ell_2 duality.□

Proof of Theorem 1.1. First suppose K(X)<\infty. On each finite cube, the adjoint of the Rademacher projection on L_2(X) is the corresponding projection on L_2(X^*). Therefore K(X^*)=K(X). By the classical characterization recalled in the introduction, both X and X^* have type strictly greater than one.

If a space Y has type p>1, with constant D, then Y^* has cotype p'=p/(p-1). Indeed, for finite tuples y_i\in Y, y_i^*\in Y^*,

Taking the supremum over the unit ball of \ell_p^m(Y) proves the claim. Apply it to Y=X and Y=X^*, and restrict the resulting cotype inequality on X^{**} to X. This direction does not use the approximation property.

Conversely, suppose that X has the approximation property and

for finite q,r\ge2 and finite constants C_F,C_Y\ge1. For an integer m, set

The parameters in (2.9) then satisfy

These positive powers of m eventually exceed any fixed multiple of (\log m)^3. Choose m once so that the hypotheses of Lemma 3.1 hold, and fix the decay sequence \rho_N given by Theorem 4.3 for m,D_g,D_f. All these parameters are independent of the finite-dimensional subspaces and approximating operators chosen below.

We first show that

For a nonzero finite-dimensional E\subset X, choose J:G\to X and \iota:E\to G by Lemma 5.1. Apply the approximation property to the compact set M=\overline{J(B_G)}. For every j\ge1, it supplies a bounded finite-rank map S_j:X\to X such that

Thus J is an operator-norm limit of finite-rank maps from G to X. Since c_m(G^*)\le D_g and c_m(X)\le D_f, the norm-limit assertion of Theorem 4.3 gives

Only the composites S_jJ converge in operator norm; no common bound on \|S_j\| is required.

Now fix any input x=(x_u)_{u\in V_N}\in L_2(V_N;X). If x=0, there is nothing to prove. Otherwise take E=\Span\{x_u:u\in V_N\} and choose the corresponding J,\iota. The family z_u=\iota(x_u) satisfies \|z\|_{L_2(G)}\le\|x\|_{L_2(X)} and Jz_u=x_u. Consequently,

which proves (5.3).

If K(X)=\infty, the characterization (1.1) supplies \lambda\ge1 such that, for every N\ge1, there are x_u\in B_X, u\in V_N, with

for all real coefficients. These vectors form an input of L_2 norm at most one, while every Walsh output satisfies

Thus w_N(\Id_X)\ge\lambda^{-1} for every N, contradicting (5.3) and \rho_N\to0. This proves K(X)<\infty.□

Proof of Corollary 1.2. If no uniform \kappa existed, choose spaces X_j with the stated fixed cotype bounds and AP, but K(X_j)\to\infty, and set

Finite-coordinate truncations converge uniformly on each compact subset of Z. Approximating its finitely many compact coordinate images by finite-rank maps therefore proves AP for Z; no uniform bound on the norms of those maps is needed. Moreover, Z^*=(\bigoplus_{j\ge1}X_j^*)_{\ell_2} isometrically. For any finite tuple z_i=(z_{ij})_j in Z, Minkowski’s inequality in \ell_{q/2} and the coordinate cotype bounds give

Thus C_q(Z)\le C, and the same argument on the dual gives C_r(Z^*)\le D. Theorem 1.1 implies K(Z)<\infty. Restriction of each finite-cube Rademacher projection to a coordinate summand gives K(X_j)\le K(Z), a contradiction.

For the entropy assertion, finite-dimensional gauge spaces have AP, so the first assertion bounds their K-constants by \kappa. The bounded-K entropy-duality theorem of Artstein–Milman–Szarek–Tomczak-Jaegermann [1, Theorem 5], in the covering-number formulation of [8, Theorem 1.1, equation (1.6), and Section 2], has constants a,b\ge1 depending only on an upper bound for the K-convexity constant of either gauge space. Polarity identifies X_{B^\circ}=X_B^*, and the finite-cube adjoint identity used above gives K(X_B^*)=K(X_B). Thus the same bound is available for the polar bodies in the reverse inequality. Also, X_{tL} is linearly isometric to X_L. Apply the covering theorem to K,tL. Since (tL)^\circ=t^{-1}L^\circ and N(t^{-1}L^\circ,cK^\circ)=N(L^\circ,ctK^\circ) for c>0, both inequalities in (1.2) follow for every t>0, with constants depending only on \kappa.□

Proof. By Gupta–Misra–Ray [6, Theorem 1.3], the pair condition implies finite cotype of F and F^*. Thus Theorem 1.1 makes F K-convex.

We use the complemented Euclidean subspace theorem of Figiel–Tomczak-Jaegermann [5], in the formulation [14, Theorem 4]: for an infinite-dimensional K-convex space F, there is D<\infty such that for every d\ge1 there are bounded maps

By the metric mapping property and the factorization

the corresponding tensor operator with \ell_2^d in place of F has norm at most CD\|A\|, uniformly in d,m,n,A.

Every algebraic tensor in E_m\otimes\ell_2 has its second coordinates in a finite-dimensional Hilbert subspace H\subset\ell_2. The inclusion of H and the orthogonal projection onto H both have norm one. The metric mapping property therefore shows that both tensor norms on tensors supported in H agree with the corresponding norms computed in \ell_2. The uniform estimate for finite-dimensional Hilbert spaces consequently holds on E_m\otimes\ell_2, and extends to its injective completion. This is (6.1) for \ell_2, with constant CD.□