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Abstract <p> We study weakly compact subsets of normed linear spaces admitting, for each <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3258_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon&gt;0\)</EquationSource> </InlineEquation>, an <i>nw</i>-continuous <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3258_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon\)</EquationSource> </InlineEquation>-selection and such that the closure of their convex hull is weakly compact in this space. Such sets are shown to be convex. An application of this result to the linear manifold of all analytic functions in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3258_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L_1\)</EquationSource> </InlineEquation> is given. </p> <p> <b> DOI</b> 10.1134/S1061920825600539 </p>

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Convexity of Weakly Compact Sets in Normed Linear Spaces

  • 期刊论文

  • 发表日期：

    2025年9月29日

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  • I.G. Tsar’kov

摘要

Abstract

We study weakly compact subsets of normed linear spaces admitting, for each \(\varepsilon>0\) , an nw-continuous \(\varepsilon\) -selection and such that the closure of their convex hull is weakly compact in this space. Such sets are shown to be convex. An application of this result to the linear manifold of all analytic functions in \(L_1\) is given.

DOI 10.1134/S1061920825600539