Displaying similar documents to “Smoothness in Banach spaces. Selected problems.”

Functions locally dependent on finitely many coordinates.

Petr Hájek, Václav Zizler (2006)

RACSAM

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The notion of functions dependent locally on finitely many coordinates plays an important role in the theory of smoothness and renormings on Banach spaces, especially when higher smoothness (C) is involved. In this note we survey most of the main results in this area, and indicate many old as well as new open problems.

On the range of the derivative of a smooth function and applications.

Robert Deville (2006)

RACSAM

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We survey recent results on the structure of the range of the derivative of a smooth real valued function f defined on a real Banach space X and of a smooth mapping F between two real Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of L(X,Y) for the existence of a Fréchet-differentiable mapping F from X into Y so that F'(X) = A. Whenever F is only assumed Gâteaux-differentiable, new phenomena appear: we discuss the existence of...

Some properties on the closed subsets in Banach spaces

Abdelhakim Maaden, Abdelkader Stouti (2006)

Archivum Mathematicum

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It is shown that under natural assumptions, there exists a linear functional does not have supremum on a closed bounded subset. That is the James Theorem for non-convex bodies. Also, a non-linear version of the Bishop-Phelps Theorem and a geometrical version of the formula of the subdifferential of the sum of two functions are obtained.

Representable Banach Spaces and Uniformly Gateaux-Smooth Norms

Frontisi, Julien (1996)

Serdica Mathematical Journal

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It is proved that a representable non-separable Banach space does not admit uniformly Gâteaux-smooth norms. This is true in particular for C(K) spaces where K is a separable non-metrizable Rosenthal compact space.

The topological complexity of sets of convex differentiable functions.

Mohammed Yahdi (1998)

Revista Matemática Complutense

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Let C(X) be the set of all convex and continuous functions on a separable infinite dimensional Banach space X, equipped with the topology of uniform convergence on bounded subsets of X. We show that the subset of all convex Fréchet-differentiable functions on X, and the subset of all (not necessarily equivalent) Fréchet-differentiable norms on X, reduce every coanalytic set, in particular they are not Borel-sets.