Results on Walsh norms, moment lifts, sign matrices and compact factorizations

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• The cotype-cotype characterization — Theorem 2.1. Assume (P) and (MP) (Pisier; not proved in the programme). A nonzero real Banach space with the approximation property is K-convex if and only if it and its dual have finite cotype.
• The decay theorem — Theorem 2.3. Under the approximation property and finite cotype of the space and its dual, the Walsh norms of the identity tend to 0 and the space contains no uniform copies of l_1^n; proved in full in the course.
• Duality of K-convexity — Lemma 3.1. K(X*) = K(X): the Rademacher projection on L_2(X*) is the adjoint of the one on L_2(X).
• K-convex spaces contain no uniform l_1^n — Proposition 3.3. K(l_1^(2^n)) >= sqrt(n/3), so a K-convex space does not contain the spaces l_1^n uniformly.
• Walsh norms — Lemma 1.1. Walsh norms lie between 2^(-n/2)|T| and |T|, are nonincreasing, invariant under amplification, and decay like 2^(-n/2) for each finite-rank operator.
• Kahane's inequality — Lemma 2.1. For Rademacher sums in a Banach space, the L_2 norm is at most 24 times the L_1 norm.
• Moment lifts — Lemma 3.2. Under fixed-length cotype bounds, bounded sign-indexed families with prescribed first moments exist.
• Chebyshev polynomials — Lemma 1.1. Coefficients of a polynomial of degree d in the Chebyshev basis are at most twice its sup norm on [-1,1], and |Q'(0)| <= d(d+1) times the sup norm.
• A coefficient of binary rank two — Lemma 2.1. An odd bounded function of a sign matrix whose row and column averages reproduce odd functions with large first-order coefficients has a Fourier coefficient of binary rank at least two, bounded below.
• Pairs of frames — Lemma 1.1. GL(n, F_2) acts transitively on pairs of frames with a prescribed pairing matrix.
• Growth from a slow decrease — Lemma 2.1. If w_(n-2m)(T) <= 2 w_n(T), then w_(2(n-2m))(T) >= c_1 w_n(T), with c_1 depending only on the cotype data.
• Uniform decay — Theorem 1.1. Walsh norms of finite-rank operators and their norm limits decay to 0 at a rate depending only on fixed-length cotype bounds.
• A compact domain — Lemma 2.1. Every finite-dimensional subspace of X factors through an injective compact map J: G -> X with |J| <= 2 and c_m(G*) <= 12 c_m(X*).
• Grothendieck pairs — Corollary 4.1. Assume (P), (GMR), (FTJ). If (E, F) is a Grothendieck pair with F infinite-dimensional with the approximation property, then (E, l_2) is a Grothendieck pair.

Reading order

Five lessons, to be read in order. The decay theorem is proved in full. The equivalence with K-convexity uses the named hypotheses (P) and (MP), two theorems of Pisier; the corollary on Grothendieck pairs also uses (GMR) and (FTJ). None of these is proved in the programme.