Isometric embeddings of separable Banach spaces into $(\ell^\infty \setminus c)\cup\{0\}$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911140232036352 |
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| author | Ribeiro, Geivison |
| author_facet | Ribeiro, Geivison |
| contents | The classical Banach--Mazur theorem asserts that every separable Banach space admits an isometric embedding into $C[0,1]$. It is also well known that every separable Banach space embeds isometrically into $\ell^\infty$. We show that such an embedding can be chosen so that its image intersects $c$ only at the origin. Moreover, we prove that any finite- or countable-dimensional, or more generally separable, subspace of $(\ell^\infty \setminus c)\cup\{0\}$ can be extended to a subspace containing an isometric copy of an arbitrary separable Banach space, while still avoiding $c$. We further establish that this extension property also holds for every subspace $D\subset \ell^\infty$ with $D\cap c=\{0\}$ and separable image in the quotient $\ell^\infty/c$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_18656 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isometric embeddings of separable Banach spaces into $(\ell^\infty \setminus c)\cup\{0\}$ Ribeiro, Geivison Functional Analysis 46B04, 46B20, 46B87, 46A16, 28A99, 15A03 The classical Banach--Mazur theorem asserts that every separable Banach space admits an isometric embedding into $C[0,1]$. It is also well known that every separable Banach space embeds isometrically into $\ell^\infty$. We show that such an embedding can be chosen so that its image intersects $c$ only at the origin. Moreover, we prove that any finite- or countable-dimensional, or more generally separable, subspace of $(\ell^\infty \setminus c)\cup\{0\}$ can be extended to a subspace containing an isometric copy of an arbitrary separable Banach space, while still avoiding $c$. We further establish that this extension property also holds for every subspace $D\subset \ell^\infty$ with $D\cap c=\{0\}$ and separable image in the quotient $\ell^\infty/c$. |
| title | Isometric embeddings of separable Banach spaces into $(\ell^\infty \setminus c)\cup\{0\}$ |
| topic | Functional Analysis 46B04, 46B20, 46B87, 46A16, 28A99, 15A03 |
| url | https://arxiv.org/abs/2508.18656 |