Locally convex spaces with the strong Gelfand-Phillips property

2021/11/10 by Тарас Банах, Banakh, Taras, Saak Gabriyelyan +1
Mathematics · #46A03 #46E10 #46E15 #54A20 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #General Topology (math.GN)

paper · pdf · doi:10.48550/arxiv.2111.05635

openalex publication_date 2021/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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⊆ X of finite scattered height such that for every functionally bounded set B⊆ X the complement B∖ 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.

Related

• The Gelfand-Phillips property for locally convex spaces
• Banach spaces with the (strong) Gelfand--Phillips property
• Gelfand--Phillips type properties of locally convex spaces
• Dual of an extended locally convex space
• The finest locally convex topology of an extended locally convex space