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Fragmentability in banach spaces: Interaction of topologies

Fragmentabilidad en espacios de Banach: interacción entre topologías

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Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

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.

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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  1. Department of Mathematics, University of Washington, Box 354350, 98195-4350, Seattle, Washington, U.S.A.

    I. Namioka

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Correspondence to I. Namioka.

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Dedicated with admiration to Professor Manuel Valdivia on the occasion of his 80th birthday

Submitted by Manuel López Pellicer

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Namioka, I. Fragmentability in banach spaces: Interaction of topologies. RACSAM 104, 283–308 (2010). https://doi.org/10.5052/RACSAM.2010.18

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