Pointwise sequential compactness and weak compactness in spaces of contents
type: Item
instance of: scholarly article
title: Pointwise sequential compactness and weak compactness in spaces of contents
zbMATH Open document ID: 0581.46037
DOI: 10.1007/BF02844885
author: Hans Weber
publication date: 1985-01-01T00:00:00Z
review text: Pointwise relative sequential compactness and sequential precompactness (sometimes called ''conditional compactness'') of sets of separable-valued contents are characterized. Using a theorem of Simon of the \(type\) ''\(<x',x_ n>\to <x',x>\) for certain x' implies weak-lim \(x_ n=x''\) a connection between pointwise convergence (in the weak topology of the range space) and weak convergence for contents with totally bounded ranges is obtained. Both together yield a criterion for weak relative sequential compactness (sequential precompactness) and, using the Eberlein-Šmulian theorem, a criterion for weak relative (countable) compactness of sets of contents, which have separable and totally bounded ranges. The assumption on the (locally convex) range space E depends only on the dual pairing \(<E,E'>\); e.g. E can be endowed with the weak as well as the strong topology of a Fréchet space.
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 46A50
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 46E27
zbMATH DE Number: 3929713
zbMATH Keywords: vector-valued measures
zbMATH Keywords: Pointwise relative sequential compactness
zbMATH Keywords: sequential precompactness
zbMATH Keywords: conditional compactness
zbMATH Keywords: separable-valued contents
zbMATH Keywords: between pointwise convergence
zbMATH Keywords: weak convergence for contents with totally bounded ranges
zbMATH Keywords: Eberlein-Šmulian theorem
zbMATH Keywords: dual pairing
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02844885
OpenAlex ID: W2044299235
published in: Rendiconti del Circolo Matematico di Palermo
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description: scientific article; zbMATH DE number 3929713