Abstract
We extend the well-known Gelfand–Phillips property for Banach spaces to locally convex spaces, defining a locally convex space E to be b-Gelfand–Phillips if every limited set in E, which is bounded in the strong topology \(\beta (E,E')\) on E, is precompact in \(\beta (E,E').\) Several characterizations of b-Gelfand–Phillips spaces are given. The problem of preservation of the b-Gelfand–Phillips property by standard operations over locally convex spaces is considered. Also we explore the b-Gelfand–Phillips property in spaces C(X) of continuous functions on a Tychonoff space X. If \(\tau\) and \({\mathcal{T}}\) are two locally convex topologies on C(X) such that \({\mathcal{T}}_p\subseteq \tau \subseteq {\mathcal{T}}\subseteq {\mathcal{T}}_k,\) where \({\mathcal{T}}_p\) is the topology of pointwise convergence and \({\mathcal{T}}_k\) is the compact-open topology on C(X), then the b-Gelfand–Phillips property of the function space \((C(X),\tau )\) implies the b-Gelfand–Phillips property of \((C(X),{\mathcal{T}}).\) If additionally X is metrizable, then the function space \(\big (C(X),{\mathcal{T}}\big )\) is b-Gelfand–Phillips.
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Banakh, T., Gabriyelyan, S. The b-Gelfand–Phillips property for locally convex spaces. Collect. Math. (2023). https://doi.org/10.1007/s13348-023-00409-5
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DOI: https://doi.org/10.1007/s13348-023-00409-5