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1
Content available remote

On the principle of local reflexivity

100%
Behrends E.
Studia Mathematica
|
1991
|
tom 100
|
nr 2
109-128
EN
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.
2
Content available remote

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

100%
Behrends E.
Studia Mathematica
|
2011
|
tom 204
|
nr 1
73-95
EN
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 $B_{ε}(f₁) ⋯ B_{ε}(fₙ)$ 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 behave in a certain admissible way when approaching {x ∈ ℝⁿ | x₁ ⋯ xₙ = 0}. We will also show that in the case of complex-valued continuous functions on [0,1] products of open subsets are always open
3
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On Bárány's theorems of Carathéodory and Helly type

100%
Behrends E.
Studia Mathematica
|
2000
|
tom 141
|
nr 3
235-250
EN
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: if $C_{n}$, n=1,2,..., are families of closed convex sets in a bounded subset of a separable Banach space X such that there exists a positive $ε_{0}$ with $⋂_{C ∈ C_{n}} (C)_{ε} = ∅$ for $ε < ε_{0}$, then there are $C_{n} ∈ C_{n}$ with $⋂_{n} (C_{n})_{ε} = ∅$ for all $ε < ε_{0}$; here $(C)_{ε}$ denotes the collection of all x with distance at most ε to C.
4
Content available remote

Metric spaces with the small ball property

64%
Behrends E., Kadets V.
Studia Mathematica
|
2001
|
tom 148
|
nr 3
275-287
EN
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 group has sbp iff it is separable and locally compact.) 3. Let B be a boundary in the bidual of an infinite-dimensional Banach space. Then B does not have sbp. In particular the set of extreme points in the unit ball of an infinite-dimensional reflexive Banach space fails to have sbp.
5
Content available remote

A selection theorem of Helly type and its applications

64%
Behrends E., Nikodem K.
Studia Mathematica
|
1995
|
tom 116
|
nr 1
43-48
EN
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.
6
Content available remote

Banach spaces which are proper M-ideals

45%
Behrends E., Harmand P.
Studia Mathematica
|
1985
|
tom 81
|
nr 2
159-169
7
Content available remote

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

44%
Behrends E.
Studia Mathematica
|
1988
|
tom 90
|
nr 2
135-151
8
Content available remote

An isomorphic Banach-Stone theorem

38%
Behrends E., Cambern M.
Studia Mathematica
|
1988
|
tom 90
|
nr 1
15-26
9
Content available remote

M-structure and the Banach-Stone theorem

32%
Behrends E., Schmidt-Bichler U.
Studia Mathematica
|
1981
|
tom 69
|
nr 1
33-40
10
Content available remote

Operators which respect norm intervals

26%
Behrends E.
Studia Mathematica
|
1981
|
tom 70
|
nr 3
203-220
11
Content available remote

$L^{p}$-Struktur in Banachräumen

26%
Behrends E.
Studia Mathematica
|
1976
|
tom 55
|
nr 1
71-85
12
Content available remote

$L^{p}$-Struktur in Banachräumen II

26%
Behrends E.
Studia Mathematica
|
1978
|
tom 62
|
nr 1
47-63
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