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Metric spaces with the small ball property

100%

Behrends E., Kadets V.

Studia Mathematica

2001

tom 148

nr 3

275-287

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.

Narrow operators and rich subspaces of Banach spaces with the Daugavet property

81%

Kadets V., Shvidkoy R., Werner D.

Studia Mathematica

2001

tom 147

nr 3

269-298

Let X be a Banach space. We introduce a formal approach which seems to be useful in the study of those properties of operators on X which depend only on the norms of the images of elements. This approach is applied to the Daugavet equation for norms of operators; in particular we develop a general theory of narrow operators and rich subspaces of spaces X with the Daugavet property previously studied in the context of the classical spaces C(K) and L₁(μ).

Ograniczanie wyników

2 Studia Mathematica

2 Kadets V.

1 Behrends E.

1 Shvidkoy R.

1 Werner D.

2 2001

Wyniki wyszukiwania

Metric spaces with the small ball property

Narrow operators and rich subspaces of Banach spaces with the Daugavet property