Analytic Approximations of Uniformly Continuous Functions in Real Banach Spaces

Uniformly Convex Functions on Banach Spaces

We study the connection between uniformly convex functions f : X → R bounded above by ‖ · ‖p, and the existence of norms on X with moduli of convexity of power type. In particular, we show that there exists a uniformly convex function f : X → R bounded above by ‖ · ‖2 if and only if X admits an equivalent norm with modulus of convexity of power type 2.

Analytic approximations of norms in separable Banach spaces

Perturbed Smooth Lipschitz Extensions of Uniformly Continuous Functions on Banach Spaces

We show that if Y is a separable subspace of a Banach space X such that both X and the quotient X/Y have Cp-smooth Lipschitz bump functions, and U is a bounded open subset of X, then, for every uniformly continuous function f : Y ∩U → R and every ε > 0, there exists a Cp-smooth Lipschitz function F : X → R such that |F (y)− f(y)| ≤ ε for every y ∈ Y ∩U . If we are given a separable subspace Y o...

Composition Operators between Weighted Banach Spaces of Analytic Functions

We characterize those analytic self-maps <p of the unit disc which generate bounded or compact composition operators Cv between given weighted Banach spaces H£° or H® of analytic functions with the weighted sup-norms. We characterize also those composition operators which are bounded or compact with respect to all reasonable weights v. 1991 Mathematics subject classification (Amer. Math. Soc): ...

Analytic Sheaves in Banach Spaces

We introduce a class of analytic sheaves in a Banach space X, that we shall call cohesive sheaves. Cohesion is meant to generalize the notion of coherence from finite dimensional analysis. Accordingly, we prove the analog of Cartan’s Theorems A and B for cohesive sheaves on pseudoconvex open subsets Ω ⊂ X, provided X has an unconditional basis.