We examine conditions under which a pair of rearrangement invariant function spaces on [0,1] or [0,∞) form a Calderón couple. A very general criterion is developed to determine whether such a pair is a Calderón couple, with numerous applications. We give, for example, a complete classification of those spaces X which form a Calderón couple with ${L}_{\infty}.$ We specialize our results to Orlicz spaces and are able to give necessary and sufficient conditions on an Orlicz function F so that the pair $({L}_{F},{L}_{\infty})$ forms a...

We construct a quasi-Banach space X which contains no basic sequence.

Let E be a Sidon subset of the integers and suppose X is a Banach space. Then Pisier has shown that E-spectral polynomials with values in X behave like Rademacher sums with respect to ${L}_{p}$-norms. We consider the situation when X is a quasi-Banach space. For general quasi-Banach spaces we show that a similar result holds if and only if E is a set of interpolation (${I}_{0}$-set). However, for certain special classes of quasi-Banach spaces we are able to prove such a result for larger sets. Thus if X is restricted...

We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X has a unique unconditional basis (up to permutation) then so does ${c}_{0}\left(X\right)$. We also give some positive results including a simpler proof that ${c}_{0}\left({\ell}_{1}\right)$ has a unique unconditional basis and a proof that ${c}_{0}\left({\ell}_{{p}_{n}}^{{N}_{n}}\right)$ has a unique unconditional basis when ${p}_{n}\uffec1$, ${N}_{n+1}\ge 2{N}_{n}$ and $({p}_{n}-{p}_{n+1})log{N}_{n}$ remains bounded.

We show that a Banach space with separable dual can be renormed to satisfy hereditarily an “almost” optimal uniform smoothness condition. The optimal condition occurs when the canonical decomposition $X***={X}^{\perp}\oplus X*$ is unconditional. Motivated by this result, we define a subspace X of a Banach space Y to be an h-ideal (resp. a u-ideal) if there is an hermitian projection P (resp. a projection P with ∥I-2P∥ = 1) on Y* with kernel ${X}^{\perp}$. We undertake a general study of h-ideals and u-ideals. For example we show that...

We show that if X is an infinite-dimensional Banach space in which every finite-dimensional subspace is λ-complemented with λ ≤ 2 then X is (1 + C√(λ-1))-isomorphic to a Hilbert space, where C is an absolute constant; this estimate (up to the constant C) is best possible. This answers a question of Kadets and Mityagin from 1973. We also investigate the finite-dimensional versions of the theorem.

We survey some questions on Rademacher series in both Banach and quasi-Banach spaces which have been the subject of extensive research from the time of Orlicz to the present day.

We show that there is no uniformly continuous selection of the quotient map $Q:{\ell}_{\infty}\to {\ell}_{\infty}/c\u2080$ relative to the unit ball. We use this to construct an answer to a problem of Benyamini and Lindenstrauss; there is a Banach space X such that there is a no Lipschitz retraction of X** onto X; in fact there is no uniformly continuous retraction from ${B}_{X**}$ onto ${B}_{X}$.

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