Preface * 1 Basic Concepts in Banach Spaces * 2 Hahn-Banach and Banach Open Mapping Theorems * 3 Weak Topologies * 4 Locally Convex Spaces * 5 Structure of Banach Spaces * 6 Schauder Bases * 7 Compact Operators on Banach Spaces * 8 Differentiability of Norms * 9 Uniform Convexity * 10 Smoothness and Structure * 11 Weakly Compactly Generated Spaces * 12 Topics in Weak Toplogy * References * Index

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Functional Analysis and Infinite-Dimensional Geometry

Using the techniques of martingale inequalities in the case of Banach space valued martingales, we give a new proof of a theorem of Enflo: every super-reflexive space admits an equivalent uniformly convex norm. Letr be a number in ]2, ∞[; we prove moreover that if a Banach spaceX is uniformly convex (resp. ifδ
 x(ɛ)/ɛ
 r whenɛ → 0) thenX admits for someq<∞ (resp. for someq<r) an equivalent norm for which the corresponding modulus of convexity satisfiesδ(ɛ)/ɛ
 q → ∞ whenɛ → 0. These results have dual analogues concerning the modulus of smoothness. Our method is to study some inequalities for martingales with values in super-reflexive or uniformly convex spaces which are characteristic of the geometry of these spaces up to isomorphism.

Martingales with values in uniformly convex spaces

Probabilistic methods in the geometry of Banach spaces

https://repository.tudelft.nl/islandora/object/uuid%3Ad8941932-86a0-43aa-b0d7-130799bbcfc1/datastream/OBJ/download

Stochastic evolution equations in UMD Banach spaces

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The distortion problem

The nonreflexive and uniformly nonoctahedral spacesX
 pgr are known to be of typep if 1≦p<2 and ρ is sufficiently large. It is shown thatX
 ρ is of type 2 if ρ>2.

Nonreflexive spaces of type 2

A relatively easy proof is given for the known theorem that a Banach space is reflexive if and only if each continuous linear functional attains its sup on the unit ball. This proof simplifies considerably for separable spaces and can be extended to give a proof that a boundedw-closed subsetX of a complete locally convex linear topological space isw-compact if and only if each continuous linear functional attains its sup onX.

Reflexivity and the sup of linear functionals

An example is given of a nonreflexive Banach space$\tilde X$ that is uniformly nonoctahedral (or uniformly non-l1(3)), in the sense that there is a λ>1 such that there is no isomorphismT ofl1(3) into$\tilde X$ for which

$$\lambda ^{ - 1} \left\| x \right\|\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \left\| {T(x)} \right\|\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \lambda \left\| x \right\|{\text{ }}if x \in l_1^{(3)}$$

.

A nonreflexive Banach space that is uniformly nonoctahedral

An example is given of a normed linear space that is not complete but for which each continuous linear functional attains its supremum on the unit ball.

A counterexample for a sup theorem in normed spaces

A Banach space has the Radon--Nikod3~m property (RNP) if it is isomorphic to a subspace of a separable du~d. Until very recently, it was thought this might be a necessary condition for RNP. The asymptotic-norming property (ANP) is introduced. It is shown that ANP is satisfied by a larger class of Banach spaces than those that are isomorphic to subspaces of separable duals, and that ANP implies RNP. For product spaces and subspaces of duals, there are significant similarities between ANP and RNP. Different formulations of ANP are studied, as well as relations between A N P and Kaded--Klee-type properties. A Banach space X is said to have the Radon--Nikodj:m property if for each finite-measure space (S, S, #) and each #-continuous, X-valued measure m with finite norm, there is a Bochner-integrable function f from S to X such that rn(E)= fEfd# if E belongs to S. A Banach space has RN P if it is isomorphic to a subspace of a separable dual ([8] and [7, Theorem 1, page 79]). Until recently, it was thought this might be a necessary condition for RNP. However, there are now examples of Banach spaces with RNP that are not isomorphic to a subspace of any separable dual ([4] and [12]). In Section 1, we will introduce ANP and show it to be a sufficient condition for RNP. Also, ANP is satisfied by all subspaces of separable duals. In Section 2, it will be shown that ANP is satisfied by some spaces not isomorphic to a subspace of any separable dual, and is preserved for certain product-type spaces. However, it remains unknown whether ANP is necessary for RNP. In Section 3, ANP will be studied for subspaces of duals. The spaces Co and LI[0, 1] do not have RNP, but /1 does have RNP. A great deal is known about conditions for RNP (e.g., see [7, pages 217 219]). Frequently,

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Robert C. James

Papers

Nonreflexive spaces of type 2

Reflexivity and the sup of linear functionals

A nonreflexive Banach space that is uniformly nonoctahedral

A counterexample for a sup theorem in normed spaces

The asymptotic-norming and Radon-Nikodým properties for Banach spaces