Free locally convex spaces with a small base

2016/04/09 by Gabriyelyan, Saak, Kakol, Jerzy
#46A03 #54A25 #54D50 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN)

paper · doi:10.48550/arxiv.1604.02555

Abstract

The paper studies the free locally convex space L(X) over a Tychonoff space X. Since for infinite X the space L(X) is never metrizable (even not Fréchet-Urysohn), a possible applicable generalized metric property for L(X) is welcome. We propose a concept (essentially weaker than first-countability) which is known under the name a \mathfrakG-base. A space X has a \em \mathfrakG-base if for every x∈ X there is a base \ U_α: α∈ℕ^ℕ\ of neighborhoods at x such that U_β⊆ U_α whenever α≤β for all α,β∈ℕ^ℕ, where α=(α(n))_n∈ℕ≤ β=(β(n))_n∈ℕ if α(n)≤β(n) for all n∈ℕ. We show that if X is an Ascoli σ-compact space, then L(X) has a \mathfrakG-base if and only if X admits an Ascoli uniformity U with a \mathfrakG-base. We prove that if X is a σ-compact Ascoli space of ℕ^ℕ-uniformly compact type, then L(X) has a \mathfrakG-base. As an application we show: (1) if X is a metrizable space, then L(X) has a \mathfrakG-base if and only if X is σ-compact, and (2) if X is a countable Ascoli space, then L(X) has a \mathfrakG-base if and only if X has a \mathfrakG-base.

Related

• On reflexivity and the Ascoli property for free locally convex spaces
• On the Ascoli property for locally convex spaces and topological groups
• \mathfrak G-bases in free (locally convex) topological vector spaces
• ω^ω-Base and infinite-dimensional compact sets in locally convex spaces
• Locally convex properties of free locally convex spaces