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On r -reflexive Banach spaces

Iryna Banakh, Taras O. Banakh, Elena Riss (2009)

Commentationes Mathematicae Universitatis Carolinae

A Banach space X is called r -reflexive if for any cover 𝒰 of X by weakly open sets there is a finite subfamily 𝒱 𝒰 covering some ball of radius 1 centered at a point x with x r . We prove that an infinite-dimensional separable Banach space X is -reflexive ( r -reflexive for some r ) if and only if each ε -net for X has an accumulation point (resp., contains a non-trivial convergent sequence) in the weak topology of X . We show that the quasireflexive James space J is r -reflexive for no r . We do not know...

Preduals of spaces of vector-valued holomorphic functions

Christopher Boyd (2003)

Czechoslovak Mathematical Journal

For U a balanced open subset of a Fréchet space E and F a dual-Banach space we introduce the topology τ γ on the space ( U , F ) of holomorphic functions from U into F . This topology allows us to construct a predual for ( ( U , F ) , τ δ ) which in turn allows us to investigate the topological structure of spaces of vector-valued holomorphic functions. In particular, we are able to give necessary and sufficient conditions for the equivalence and compatibility of various topologies on spaces of vector-valued holomorphic functions....

Reflexivity of inductive limits

Jan Kučera (2004)

Czechoslovak Mathematical Journal

An inductive locally convex limit of reflexive topological spaces is reflexive iff it is almost regular.

Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces

Victor Klee, Libor Veselý, Clemente Zanco (1996)

Studia Mathematica

For combining two convex bodies C and D to produce a third body, two of the most important ways are the operation ∓ of forming the closure of the vector sum C+D and the operation γ̅ of forming the closure of the convex hull of C ⋃ D. When the containing normed linear space X is reflexive, it follows from weak compactness that the vector sum and the convex hull are already closed, and from this it follows that the class of all rotund bodies in X is stable with respect to the operation ∓ and the class...

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