Locally convex spaces with the strong Gelfand-Phillips property

Taras Banakh, Saak Gabriyelyan

Research output: Working paper/PreprintPreprint

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Abstract

We introduce the strong Gelfand-Phillips property for locally convex spaces and give several characterizations of this property. We characterize the strong Gelfand-Phillips property among locally convex spaces admitting a stronger Banach space topology. If $C_{\mathcal T}(X)$ is a space of continuous functions on a Tychonoff space $X$, endowed with a locally convex topology $\mathcal T$ between the pointwise topology and the compact-open topology, then: (a) the space $C_{\mathcal T}(X)$ has the strong Gelfand-Phillips property iff $X$ contains a compact countable subspace $K\subseteq X$ of finite scattered height such that for every functionally bounded set $B\subseteq X$ the complement $B\setminus K$ is finite, (b) the subspace $C^b_{\mathcal T}(X)$ of $C_{\mathcal T}(X)$ consisting of all bounded functions on $X$ has the strong Gelfand-Phillips property iff $X$ is a compact countable space of finite scattered height.
Original languageEnglish
StatePublished - 10 Nov 2021

Keywords

  • math.FA
  • math.GN
  • 46A03, 46E10, 46E15, 54A20

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