On the Ascoli property for locally convex spaces

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13 Scopus citations


We characterize Ascoli spaces by showing that a Tychonoff space X is Ascoli iff the canonical map from the free locally convex space L(X) over X into Ck(Ck(X)) is an embedding of locally convex spaces. We prove that an uncountable direct sum of non-trivial locally convex spaces is not Ascoli. If a c0-barrelled space E is weakly Ascoli, then E is linearly isomorphic to a dense subspace of RΓ for some set Γ. Consequently, a Fréchet space E is weakly Ascoli iff E=RN for some N≤ω. If X is a μ-space and a kR-space (for example, metrizable), then Ck(X) is weakly Ascoli iff X is discrete. If X is a μ-space, then the space Mc(X) of all regular Borel measures on X with compact support is Ascoli in the weak topology iff X is finite. The weak dual space of a metrizable barrelled space E is Ascoli iff E is finite-dimensional.

Original languageEnglish
Pages (from-to)517-530
Number of pages14
JournalTopology and its Applications
StatePublished - 1 Oct 2017


  • Banach space
  • Barrelled space
  • Direct sum of locally convex spaces
  • Free locally convex space
  • Function space
  • The Ascoli property

ASJC Scopus subject areas

  • Geometry and Topology


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