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Abstract <p> We study boundedly weakly compact sets admitting, for each <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3067_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon&gt;0\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3067_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathrm{nw}\)</EquationSource> </InlineEquation>-continuous (norm-weak continuous) <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3067_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon\)</EquationSource> </InlineEquation>-selections of the near-best metric projection operator. Such a set in a reflexive Kadec–Klee space is shown to be a sun. For an approximatively compact set it is verified that this set admits an <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3067_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathrm{nw}\)</EquationSource> </InlineEquation>-continuous <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3067_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon\)</EquationSource> </InlineEquation>-selection for each <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3067_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon&gt;0\)</EquationSource> </InlineEquation> if and only if it admits an <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3067_Article_IEq7.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathrm{nn}\)</EquationSource> </InlineEquation>-continuous (norm-norm continuous) <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3067_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon\)</EquationSource> </InlineEquation>-selection for each <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3067_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon&gt;0\)</EquationSource> </InlineEquation>. </p>

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Solarity of Boundedly Weakly Compact Sets

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • I. G. Tsar’kov

摘要

Abstract

We study boundedly weakly compact sets admitting, for each \(\varepsilon>0\) , \(\mathrm{nw}\) -continuous (norm-weak continuous) \(\varepsilon\) -selections of the near-best metric projection operator. Such a set in a reflexive Kadec–Klee space is shown to be a sun. For an approximatively compact set it is verified that this set admits an \(\mathrm{nw}\) -continuous \(\varepsilon\) -selection for each \(\varepsilon>0\) if and only if it admits an \(\mathrm{nn}\) -continuous (norm-norm continuous) \(\varepsilon\) -selection for each \(\varepsilon>0\) .