Weak^*-weak points of continuity and very smooth points in Bochner function spaces L^p(μ, X)

Abstract

Suppose that ([0,1],𝒜,μ) is the Lebesgue measure space and 1< p<∞ with 1/p+1/q=1. We prove that an unit vector h∈ L^p(μ, X)^* is a weak^*-weak point of continuity of S(L^p(μ, X)^*) if and only if h(t)/|h(t)| is a weak^*-weak point of continuity of S(X^*) for almost every t∈ [0, 1] with h(t)≠ 0, where X is a Banach space such that X^* is separable. As an application of this result, under the same hypothesis on X, we prove that a point f∈ L^p(μ, X)₁ is a very smooth point if and only if f(t)/|f(t)|∈ S(X) is a very smooth point for almost every t∈ [0, 1] with f(t)≠ 0. We give several examples to show that these results does not hold for p=1.

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BibTeX RIS

BibTeX

@misc{dwivedi2026weakweak,
  title = {{Weak$^*$-weak points of continuity and very smooth points in Bochner function spaces $L^p(μ, X)$}},
  author = {Saurabh Dwivedi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23502},
  eprint = {2610.23502},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23502},
}

RIS

TY  - UNPB
TI  - Weak^*-weak points of continuity and very smooth points in Bochner function spaces L^p(μ, X)
AU  - Saurabh Dwivedi
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23502
UR  - https://arxiv.org/abs/2610.23502
AB  - Suppose that ([0,1],𝒜,μ) is the Lebesgue measure space and 1< p<∞ with 1/p+1/q=1. We prove that an unit vector h∈ L^p(μ, X)^* is a weak^*-weak point of continuity of S(L^p(μ, X)^*) if and only if h(t)/|h(t)| is a weak^*-weak point of continuity of S(X^*) for almost every t∈ [0, 1] with h(t)≠ 0, where X is a Banach space such that X^* is separable. As an application of this result, under the same hypothesis on X, we prove that a point f∈ L^p(μ, X)_1 is a very smooth point if and only if f(t)/|f(t)|∈ S(X) is a very smooth point for almost every t∈ [0, 1] with f(t)≠ 0. We give several examples to show that these results does not hold for p=1.
ER  -

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