On weak*-basic sequences in duals and biduals of spaces C(X) and Quojections
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| Format: | Preprint |
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2026
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| _version_ | 1866908792262754304 |
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| author | Kakol, Jerzy Lopez-Pellicer, Manuel Sliwa, Wieslaw |
| author_facet | Kakol, Jerzy Lopez-Pellicer, Manuel Sliwa, Wieslaw |
| contents | We show that for infinite Tychonoff spaces X and Y the weak*-dual of Ck(X x Y) contains a basic sequence; moreover, the weak*-bidual of Ck(X) contains such a sequence as well. When X and Y are infinite compact spaces, we single out a concrete sequence (μn) of finitely supported signed measures on X x Y with quantitative small-rectangle estimates, and we prove that every subsequence of (μn) admits a further subsequence which is strongly normal and forms a weak*-basic sequence in the dual C(X x Y)* of the Banach space C(X x Y). We also study the weak*-basic sequence problem for Frechet locally convex spaces in the class of quojections, and prove that for every quojection E the bidual E** admits a weak*-basic sequence, while a long-standing open problem asks whether the dual of every infinite-dimensional Banach space admits a basic sequence in the weak*-topology. Several examples and open questions are included, in particular for spaces C(X) and for inductive limits of Frechet spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_19873 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On weak*-basic sequences in duals and biduals of spaces C(X) and Quojections Kakol, Jerzy Lopez-Pellicer, Manuel Sliwa, Wieslaw Functional Analysis General Topology 54C35, 54G12, 54H05, 46A03 We show that for infinite Tychonoff spaces X and Y the weak*-dual of Ck(X x Y) contains a basic sequence; moreover, the weak*-bidual of Ck(X) contains such a sequence as well. When X and Y are infinite compact spaces, we single out a concrete sequence (μn) of finitely supported signed measures on X x Y with quantitative small-rectangle estimates, and we prove that every subsequence of (μn) admits a further subsequence which is strongly normal and forms a weak*-basic sequence in the dual C(X x Y)* of the Banach space C(X x Y). We also study the weak*-basic sequence problem for Frechet locally convex spaces in the class of quojections, and prove that for every quojection E the bidual E** admits a weak*-basic sequence, while a long-standing open problem asks whether the dual of every infinite-dimensional Banach space admits a basic sequence in the weak*-topology. Several examples and open questions are included, in particular for spaces C(X) and for inductive limits of Frechet spaces. |
| title | On weak*-basic sequences in duals and biduals of spaces C(X) and Quojections |
| topic | Functional Analysis General Topology 54C35, 54G12, 54H05, 46A03 |
| url | https://arxiv.org/abs/2601.19873 |