Unique Hahn-Banach extensions in Bochner L^p spaces

Abstract

Let Y be a closed subspace of a real Banach space X. Let (Ω,Σ,μ) be a complete finite measure space and let 1<p<∞. We prove that Y has property-(U), property-(SU), property-(wU) in X if and only if L^p(μ,Y) has the corresponding property in L^p(μ,X). No Radon-Nikodým assumption on X^* is required. The proofs use w^*-measurable representations of dual functionals and a simultaneous scalar-lifting construction. A norm-minimal representation for an element in L^p(μ,X)^* is found suitable in this case. We also study the point of continuity of the identity map under different topologies on the unit sphere of L^p(μ,X)^*. For Bochner-integrable functionals, the pointwise w^*−w-continuity yields the uniqueness of the extension to L^p(μ,X^**). Examples show why arbitrary w^*-measurable representatives cannot replace the norm-minimal representatives in this context.

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BibTeX RIS

BibTeX

@misc{karak2026unique,
  title = {{Unique Hahn-Banach extensions in Bochner $L^p$ spaces}},
  author = {Sainik Karak and Tanmoy Paul},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03203},
  eprint = {2611.03203},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03203},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Unique Hahn-Banach extensions in Bochner L^p spaces
AU  - Sainik Karak
AU  - Tanmoy Paul
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03203
UR  - https://arxiv.org/abs/2611.03203
AB  - Let Y be a closed subspace of a real Banach space X. Let (Ω,Σ,μ) be a complete finite measure space and let 1<p<∞. We prove that Y has property-(U), property-(SU), property-(wU) in X if and only if L^p(μ,Y) has the corresponding property in L^p(μ,X). No Radon-Nikodým assumption on X^* is required. The proofs use w^*-measurable representations of dual functionals and a simultaneous scalar-lifting construction. A norm-minimal representation for an element in L^p(μ,X)^* is found suitable in this case. We also study the point of continuity of the identity map under different topologies on the unit sphere of L^p(μ,X)^*. For Bochner-integrable functionals, the pointwise w^*−w-continuity yields the uniqueness of the extension to L^p(μ,X^**). Examples show why arbitrary w^*-measurable representatives cannot replace the norm-minimal representatives in this context.
ER  -

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