Structural Hierarchy of Reid Class of non-Archimedean Banach Spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913009459265536 |
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| author | Mihara, Tomoki |
| author_facet | Mihara, Tomoki |
| contents | Let $k$ be a complete valuation field. We formulate a class $\mathscr{R}$ of Banach $k$-vector spaces analogous to Reid class of Abelian groups. We formulate an analogue of the hierarchy of Reid class introduced by K.\ Eda, and verify a counterpart of the classification theorem of Reid class by K.\ Eda. As an application, we verify that the Banach $\mathbb{C}_p$-vector spaces \begin{eqnarray*} & & \ell^{\infty}(\mathbb{N},\mathbb{C}_p), \text{\rm C}_0(\mathbb{N},\mathbb{C}_p), \ell^{\infty}(\mathbb{N},\text{\rm C}_0(\mathbb{N},\mathbb{C}_p)), \text{\rm C}_0(\mathbb{N},\ell^{\infty}(\mathbb{N},\mathbb{C}_p)), \\ & & \ell^{\infty}(\mathbb{N},\text{\rm C}_0(\mathbb{N},\ell^{\infty}(\mathbb{N},\mathbb{C}_p))), \text{\rm C}_0(\mathbb{N},\ell^{\infty}(\mathbb{N},\text{\rm C}_0(\mathbb{N},\mathbb{C}_p))), \end{eqnarray*} and so on are all distinct, the Banach $\mathbb{C}_p$-vector space of bounded continuous functions $\mathbb{Q} \to \mathbb{C}_p$ and its dual Banach $\mathbb{C}_p$-vector spaces cannot be expressed by iterated application of bounded direct product and completed direct sum, and there is no left adjoint functor of the forgetful functor from $\mathscr{R}$ to the category of Banach $\mathbb{C}_p$-vector spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05330 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Structural Hierarchy of Reid Class of non-Archimedean Banach Spaces Mihara, Tomoki Logic Functional Analysis Number Theory Let $k$ be a complete valuation field. We formulate a class $\mathscr{R}$ of Banach $k$-vector spaces analogous to Reid class of Abelian groups. We formulate an analogue of the hierarchy of Reid class introduced by K.\ Eda, and verify a counterpart of the classification theorem of Reid class by K.\ Eda. As an application, we verify that the Banach $\mathbb{C}_p$-vector spaces \begin{eqnarray*} & & \ell^{\infty}(\mathbb{N},\mathbb{C}_p), \text{\rm C}_0(\mathbb{N},\mathbb{C}_p), \ell^{\infty}(\mathbb{N},\text{\rm C}_0(\mathbb{N},\mathbb{C}_p)), \text{\rm C}_0(\mathbb{N},\ell^{\infty}(\mathbb{N},\mathbb{C}_p)), \\ & & \ell^{\infty}(\mathbb{N},\text{\rm C}_0(\mathbb{N},\ell^{\infty}(\mathbb{N},\mathbb{C}_p))), \text{\rm C}_0(\mathbb{N},\ell^{\infty}(\mathbb{N},\text{\rm C}_0(\mathbb{N},\mathbb{C}_p))), \end{eqnarray*} and so on are all distinct, the Banach $\mathbb{C}_p$-vector space of bounded continuous functions $\mathbb{Q} \to \mathbb{C}_p$ and its dual Banach $\mathbb{C}_p$-vector spaces cannot be expressed by iterated application of bounded direct product and completed direct sum, and there is no left adjoint functor of the forgetful functor from $\mathscr{R}$ to the category of Banach $\mathbb{C}_p$-vector spaces. |
| title | Structural Hierarchy of Reid Class of non-Archimedean Banach Spaces |
| topic | Logic Functional Analysis Number Theory |
| url | https://arxiv.org/abs/2604.05330 |