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      \(\omega^\omega\)-Base and infinite-dimensional compact sets in locally convex spaces

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          Abstract

          A locally convex space (lcs) \(E\) is said to have an \(\omega^{\omega}\)-base if \(E\) has a neighborhood base \(\{U_{\alpha}:\alpha\in\omega^\omega\}\) at zero such that \(U_{\beta}\subseteq U_{\alpha}\) for all \(\alpha\leq\beta\). The class of lcs with an \(\omega^{\omega}\)-base is large, among others contains all \((LM)\)-spaces (hence \((LF)\)-spaces), strong duals of distinguished Fr\'echet lcs (hence spaces of distributions \(D'(\Omega)\)). A remarkable result of Cascales-Orihuela states that every compact set in a lcs with an \(\omega^{\omega}\)-base is metrizable. Our main result shows that every uncountable-dimensional lcs with an \(\omega^{\omega}\)-base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional space \(\varphi\) endowed with the finest locally convex topology has an \(\omega^\omega\)-base but contains no infinite-dimensional compact subsets. It turns out that \(\varphi\) is a unique infinite-dimensional locally convex space which is a \(k_{\mathbb{R}}\)-space containing no infinite-dimensional compact subsets. Applications to spaces \(C_{p}(X)\) are provided.

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          Author and article information

          Journal
          08 July 2020
          Article
          2007.04420
          c7d16d3b-f24e-43fa-a985-f87157351300

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          math.GN

          Geometry & Topology
          Geometry & Topology

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