Reflexive extended locally convex spaces

TL;DR

This paper investigates the reflexivity of extended locally convex spaces using the topology of uniform convergence on bounded sets, establishing conditions under which reflexivity is preserved in subspaces and extended normed spaces.

Contribution

It introduces the use of the topology b to analyze reflexivity in extended locally convex spaces, showing reflexivity is a three-space property for extended normed spaces.

Findings

Reflexivity of elcs is equivalent to all open subspaces being reflexive.

Reflexivity is a three-space property in extended normed spaces.

The topology b is effective for studying reflexivity in these spaces.

Abstract

For an extended locally convex space (elcs) $(X,\tau)$, the authors in [10] studied the topology $\tau_{ucb}$ of uniform convergence on bounded subsets of $(X,\tau)$ on the dual $X^*$ of $(X,\tau)$. In the present paper, we use the topology $\tau_{ucb}$ to explore the reflexive property of extended locally convex spaces. It is shown that an elcs is (semi) reflexive if and only if any of its open subspaces is (semi) reflexive. For an extended normed space, we show that reflexivity is a three-space property.