Valdivia's lifting theorem for non-metrizable spaces. Preprint

TL;DR

This paper extends Valdivia's lifting theorem to a broader class of range spaces with a sequential web structure, including non-metrizable and non-barrelled spaces, with applications to distribution spaces.

Contribution

It generalizes Valdivia's lifting theorem to include non-metrizable and non-barrelled spaces with sequential web structures, expanding its applicability.

Findings

Extended the lifting theorem to non-metrizable spaces

Provided examples including distribution spaces with wavefront sets

Showed wider applicability of the closed graph theorem

Abstract

The lifting theorem of Valdivia concerning (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 metrizable, nor strictly barrelled, and the range space need not even be barrelled. Distinguishing examples are provided that include natural constructions of range spaces connected with applications, such as the space of distributions having their wavefront sets in a specified closed cone. The same and other examples could also serve as domain spaces for the closed graph theorem of Valdivia, revealing a much wider collection of domain spaces that can be used in that result.