PreprintCC BY 4.0arXiv
Pointwise function spaces over compacta are not weak Banach spaces
Abstract
Let K be a compact Hausdorff space and let E be an infinite-dimensional real Banach space. We prove that there is no continuous bijection h: Cp(K)→ Ew whose inverse is continuous at h(0). Consequently, Cp(K) and Cw(L) are not homeomorphic for any infinite compact Hausdorff spaces K and L. This settles Krupski's problem and its two-space version due to Krupski and Marciszewski, and answers a question of Kąkol, Leiderman, and Michalak concerning Cp([0,1]) and weak Banach spaces.
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BibTeX
@misc{kania2026pointwise,
title = {{Pointwise function spaces over compacta are not weak Banach spaces}},
author = {Tomasz Kania and Jerzy Kąkol},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.17549},
eprint = {2608.17549},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.17549},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Pointwise function spaces over compacta are not weak Banach spaces
AU - Tomasz Kania
AU - Jerzy Kąkol
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2608.17549
UR - https://arxiv.org/abs/2608.17549
AB - Let K be a compact Hausdorff space and let E be an infinite-dimensional real Banach space. We prove that there is no continuous bijection h: C_p(K)→ E_w whose inverse is continuous at h(0). Consequently, C_p(K) and C_w(L) are not homeomorphic for any infinite compact Hausdorff spaces K and L. This settles Krupski's problem and its two-space version due to Krupski and Marciszewski, and answers a question of Kąkol, Leiderman, and Michalak concerning C_p([0,1]) and weak Banach spaces.
ER -