Fragmentability in banach spaces: Interaction of topologies

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Abstract

Let (X, τ) be a topological space and let ρ be a metric on X. Then one occasionally encounters the following situation: For each ε > 0 and a non-empty subset A ⊂ X, there exists a τ-open subset U of X such that U ∩ A = ϕ and ρ-diameter of U ∩ A is less than ∈. If this is the case then (X, \gt) is said to be fragmented by \gr. For instance a weakly compact subset of a Banach space with the weak topology is fragmented by the norm metric, and this fact has many consequences. For non-compact spaces, the natural analog of fragmentability is \gs-fragmentability. In this exposition, these two notions are examined and their applications are described.

Keywords

fragmentability σ-fragmentability renorming 

Mathematics Subject Classifications

(2000 MSC) 54E99 46B22 

Fragmentabilidad en espacios de Banach: interacción entre topologías

Resumen

Cuando en un espacio topológico (X, τ) tenemos además una métrica ρ, a veces la siguiente situación se presenta: para cada ∈ > 0 y para cada conjunto A ⊂ X, existe un subconjunto τ-abierto U de X tal que U ∩ A = ϕ y ρ-diámetro de U ∩ A es menor que ∈. Si este último es el caso, (X, τ) se dice que está fragmentado por ρ. Por ejemplo, los subconjuntos débilmente compactos de un espacio de Banach con su topología débil están fragmentados por la métrica asociada a la norma: este resultado tiene muchas consecuencias. Para espacios que no son compactos, el análogo natural de la noción de fragmentabilidad es la noción σ-fragmentabilidad. En este artículo expositivo, analizamos las nociones de fragmentabilidad y σ-fragmentabilidad así como aplicaciones de las mismas.

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Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleU.S.A.

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