Locally convex spaces with hemicompact weak* dual
Abstract
For a locally convex space L let L'σ denote its dual endowed with the weak* topology σ(L',L). We prove that L'σ is hemicompact if and only if L with the Mackey topology τ(L,L') is metrizable and barrelled, and if and only if L'σ is σ-compact and quasi-complete. The key step is that a hemicompact L'σ is quasi-complete. The Mackey topology τ(L,L') is metrizable if and only if L' is a countable union of absolutely convex weak*-compact sets; spaces with normable barrelled Mackey topology are characterized similarly. As a consequence, a metrizable (in particular, normed) space L is barrelled if and only if L'σ is hemicompact. For function spaces we obtain: (Ck(X))'σ is hemicompact if and only if X is hemicompact, and (Cp(X))'σ is hemicompact if and only if X is countable and all compact subsets of X are finite.
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BibTeX
@misc{reznichenko2026locally,
title = {{Locally convex spaces with hemicompact weak* dual}},
author = {Evgenii Reznichenko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24615},
eprint = {2610.24615},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24615},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Locally convex spaces with hemicompact weak* dual
AU - Evgenii Reznichenko
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24615
UR - https://arxiv.org/abs/2610.24615
AB - For a locally convex space L let L'_σ denote its dual endowed with the weak* topology σ(L',L). We prove that L'_σ is hemicompact if and only if L with the Mackey topology τ(L,L') is metrizable and barrelled, and if and only if L'_σ is σ-compact and quasi-complete. The key step is that a hemicompact L'_σ is quasi-complete. The Mackey topology τ(L,L') is metrizable if and only if L' is a countable union of absolutely convex weak*-compact sets; spaces with normable barrelled Mackey topology are characterized similarly. As a consequence, a metrizable (in particular, normed) space L is barrelled if and only if L'_σ is hemicompact. For function spaces we obtain: (C_k(X))'_σ is hemicompact if and only if X is hemicompact, and (C_p(X))'_σ is hemicompact if and only if X is countable and all compact subsets of X are finite.
ER -