Mathematical Analysis and its Contemporary Applications

Norm-attainment in locally convex spaces: Weak-$^*$ topology, inductive limits, and reflexivity.

10.30495/maca.2025.2057641.1134
Volume 7, Issue 4
Autumn 2025
Pages 13-25
Mathematical Analysis and its Contemporary Applications

Document Type : Original Article

Authors

Evans N. Mogi
Priscah Moraa Ohuru

Jaramogi Oginga Odinga University of Science and Technology, Kenya

Abstract
We characterize norm-attaining functionals in locally convex spaces (LCS), with particular focus on three fundamental aspects: the weak-$^*$ (weak-star) topology in dual spaces, inductive limits (including LF-spaces and DF-spaces), and reflexivity conditions. Our main results establish that (1) norm-attainment in the weak-$^*$ dual coincides precisely with the canonical embedding $X \hookrightarrow X^{**}$; (2) strict inductive limits (such as $\mathcal{D}(\mathbb{R})$) permit non-attaining functionals, whereas Montel spaces ensure universal attainment; and (3) both barrelledness and reflexivity conditions recover norm-attainment through weak-$^*$ continuity. This work extends classical Banach space techniques to general LCS settings, revealing the crucial interplay between compactness properties and approximation methods in determining norm-attainment behavior.

Keywords

Norm-attainment
Inductive limits
Montel spaces
Reflexivity
Dual spaces
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  • Receive Date 11 April 2025
  • Accept Date 09 June 2025

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