Valdivia's lifting theorem for non-metrizable spaces

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Abstract

Valdivia's lifting theorem of (pre) compact sets and convergent (respectively, Cauchy) sequences from a quasi-(LB) space to a metrizable, strictly barrelled space is extended to a strictly larger collection of range spaces. Specifically, we assume that the range space has a sequential web structure and do not require that it be strictly barrelled, and need not even be barrelled. Distinguishing examples are provided that include natural constructions of range spaces connected with applications, such as DΓ, the space of distributions having their wavefront sets in a specified closed cone Γ. We observe that the same and other examples could also serve as domain spaces for Valdivia's closed graph theorem, revealing a much wider collection of domain spaces that can be used in that result.

Original languageEnglish
Article number108160
JournalTopology and its Applications
Volume317
DOIs
StatePublished - Aug 1 2022

Keywords

  • Closed graph theorem
  • Lifting theorem
  • Quasi-(LB) space
  • Strictly barrelled
  • Web

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