On the product of Weak Asplund locally convex spaces
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913610750492672 |
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| author | Kakol, Jerzy Leiderman, Arkady |
| author_facet | Kakol, Jerzy Leiderman, Arkady |
| contents | For locally convex spaces, we systematize several known equivalent definitions of Fréchet (G\^ ateaux) Differentiability Spaces and Asplund (Weak Asplund) Spaces.
As an application, we extend the classical Mazur's theorem as follows: Let $E$ be a separable Baire locally convex space and let $Y$ be the product $\prod_{α\in A} E_α$ of any family of separable Fréchet spaces; then the product $E \times Y$ is Weak Asplund. Also, we prove that the product $Y$ of any family of Banach spaces $(E_α)$ is an Asplund locally convex space if and only if each $E_α$ is Asplund.
Analogues of both results are valid under the same assumptions, if $Y$ is the $Σ$-product of any family $(E_α)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10221 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the product of Weak Asplund locally convex spaces Kakol, Jerzy Leiderman, Arkady Functional Analysis General Topology Primary 46A04, Secondary 54B10, 54E52 For locally convex spaces, we systematize several known equivalent definitions of Fréchet (G\^ ateaux) Differentiability Spaces and Asplund (Weak Asplund) Spaces. As an application, we extend the classical Mazur's theorem as follows: Let $E$ be a separable Baire locally convex space and let $Y$ be the product $\prod_{α\in A} E_α$ of any family of separable Fréchet spaces; then the product $E \times Y$ is Weak Asplund. Also, we prove that the product $Y$ of any family of Banach spaces $(E_α)$ is an Asplund locally convex space if and only if each $E_α$ is Asplund. Analogues of both results are valid under the same assumptions, if $Y$ is the $Σ$-product of any family $(E_α)$. |
| title | On the product of Weak Asplund locally convex spaces |
| topic | Functional Analysis General Topology Primary 46A04, Secondary 54B10, 54E52 |
| url | https://arxiv.org/abs/2412.10221 |