EUDML

We prove a version of the local reflexivity theorem which is, in a sense, the most general one: our main theorem characterizes the conditions which can be imposed additionally on the usual local reflexivity map provided that these conditions are of a certain general type. It is then shown how known and new local reflexivity theorems can be derived. In particular, the compatibility of the local reflexivity map with subspaces and operators is investigated.

$L^{p}$ -Struktur in Banachräumen

Ehrhard Behrends — 1976

Studia Mathematica

$L^{p}$ -Struktur in Banachräumen II

Ehrhard Behrends — 1978

Studia Mathematica

On Bárány's theorems of Carathéodory and Helly type

Ehrhard Behrends — 2000

Studia Mathematica

The paper begins with a self-contained and short development of Bárány’s theorems of Carathéodory and Helly type in finite-dimensional spaces together with some new variants. In the second half the possible generalizations of these results to arbitrary Banach spaces are investigated. The Carathéodory-Bárány theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Bárány theorem reads as follows:...

On the geometry of spaces of $C_{0}$ K-valued operators

Ehrhard Behrends — 1988

Studia Mathematica

Operators which respect norm intervals

Ehrhard Behrends — 1981

Studia Mathematica

Products of n open subsets in the space of continuous functions on [0,1]

Ehrhard Behrends — 2011

Studia Mathematica

Let O₁,...,Oₙ be open sets in C[0,1], the space of real-valued continuous functions on [0,1]. The product O₁ ⋯ Oₙ will in general not be open, and in order to understand when this can happen we study the following problem: given f₁,..., fₙ ∈ C[0,1], when is it true that f₁ ⋯ fₙ lies in the interior of $B_{ε} (f ₁) \dots B_{ε} (f ₙ)$ for all ε > 0 ? ( $B_{ε}$ denotes the closed ball with radius ε and centre f.) The main result of this paper is a characterization in terms of the walk t ↦ γ(t): = (f₁(t),..., fₙ(t)) in ℝⁿ. It has to...

A selection theorem of Helly type and its applications

Ehrhard Behrends; Kazimierz Nikodem — 1995

Studia Mathematica

We prove an abstract selection theorem for set-valued mappings with compact convex values in a normed space. Some special cases of this result as well as its applications to separation theory and Hyers-Ulam stability of affine functions are also given.

An isomorphic Banach-Stone theorem

Ehrhard Behrends; Michael Cambern — 1988

Studia Mathematica

Banach spaces which are proper M-ideals

Ehrhard Behrends; Peter Harmand — 1985

Studia Mathematica

Strongly mixing sequences of measure preserving transformations

Ehrhard Behrends; Jörg Schmeling — 2001

Czechoslovak Mathematical Journal

We call a sequence $(T_{n})$ of measure preserving transformations strongly mixing if $P (T_{n}^{- 1} A \cap B)$ tends to $P (A) P (B)$ for arbitrary measurable $A$ , $B$ . We investigate whether one can pass to a suitable subsequence $(T_{n_{k}})$ such that $\frac{1}{K} \sum_{k = 1}^{K} f (T_{n_{k}}) ⟶ \int f d P$ almost surely for all (or “many”) integrable $f$ .

M-structure and the Banach-Stone theorem

Ehrhard Behrends; Ursula Schmidt-Bichler — 1981

Studia Mathematica

Metric spaces with the small ball property

Ehrhard Behrends; Vladimir M. Kadets — 2001

Studia Mathematica

A metric space (M,d) is said to have the small ball property (sbp) if for every ε₀ > 0 it is possible to write M as the union of a sequence (B(xₙ,rₙ)) of closed balls such that the rₙ are smaller than ε₀ and lim rₙ = 0. We study permanence properties and examples of sbp. The main results of this paper are the following: 1. Bounded convex closed sets in Banach spaces have sbp only if they are compact. 2. Precisely the finite-dimensional Banach spaces have sbp. (More generally: a complete metric...