Bohr compactification of separable locally convex spaces

Sidney A. Morris; Sidney A. Morris

doi:10.3934/era.2025060

Electronic Research Archive

2025, Volume 33, Issue 3: 1333-1336. doi: 10.3934/era.2025060

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Expository

Bohr compactification of separable locally convex spaces

Sidney A. Morris ^{1,2
,
,}

1.
Department of Mathematical and Physical Sciences, La Trobe University, Melbourne, Victoria 3086, Australia
2.
School of Engineering, IT and Physical Sciences, Federation University Australia, PO Box 663, Ballarat, Victoria 3353, Australia

Received: 22 December 2024 Revised: 20 February 2025 Accepted: 21 February 2025 Published: 05 March 2025

It is not well-known that for each separable real locally convex space, its Bohr compactification is isomorphic as a topological group to the Bohr compactification of the topological group $ \mathbb{R} $ of all real numbers. This is the case for each separable real Banach space. In this expository research note, we state the results explicitly and provide accessible proofs.
- Bohr compactification,
- abelian topological group,
- Banach space,
- Pontryagin duality,
- dual group,
- separable locally convex space
Citation: Sidney A. Morris. Bohr compactification of separable locally convex spaces[J]. Electronic Research Archive, 2025, 33(3): 1333-1336. doi: 10.3934/era.2025060

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Abstract

It is not well-known that for each separable real locally convex space, its Bohr compactification is isomorphic as a topological group to the Bohr compactification of the topological group $ \mathbb{R} $ of all real numbers. This is the case for each separable real Banach space. In this expository research note, we state the results explicitly and provide accessible proofs.

References

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