Position of $L(X, Y)$ in $Lip_0(X, Y)$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913889149517824 |
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| author | Karn, Anil Kumar Mandal, Arindam |
| author_facet | Karn, Anil Kumar Mandal, Arindam |
| contents | We prove that $L(X,Y)$ is complemented in $Lip_0(X, Y)$ (via a norm-one projection) provided that $Y$ is a dual space. Next, we introduce a vector-valued Lipschitz-free space $F_Y(X)$, a linear contraction $β_X^Y: F_Y(X) \to Y$ and prove that the quotient space ${Lip_{0}(X, Y)}/{L(X, Y) }$ is isometrically isomorphic to $L(\ker(β_X^Y), Y)$ whenever $Y$ is injective. We also consider a $Y$-valued duality pairing between $Lip_0(X, Y)$ and $F_Y(X)$ and obtain a necessary and sufficient condition for a Lipschitz map to be linear. As an application, we describe $\ker(β_{\mathbb{R}}^{\mathbb{R}})$ as the pre-dual of the quotient space $L^{\infty}(\mathbb{R})/\tilde{\mathbb{R}}$ where $\tilde{\mathbb{R}}$ is the set of all constant maps on $\mathbb{R}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09324 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Position of $L(X, Y)$ in $Lip_0(X, Y)$ Karn, Anil Kumar Mandal, Arindam Functional Analysis Primary 46B20, Secondary 46B03 We prove that $L(X,Y)$ is complemented in $Lip_0(X, Y)$ (via a norm-one projection) provided that $Y$ is a dual space. Next, we introduce a vector-valued Lipschitz-free space $F_Y(X)$, a linear contraction $β_X^Y: F_Y(X) \to Y$ and prove that the quotient space ${Lip_{0}(X, Y)}/{L(X, Y) }$ is isometrically isomorphic to $L(\ker(β_X^Y), Y)$ whenever $Y$ is injective. We also consider a $Y$-valued duality pairing between $Lip_0(X, Y)$ and $F_Y(X)$ and obtain a necessary and sufficient condition for a Lipschitz map to be linear. As an application, we describe $\ker(β_{\mathbb{R}}^{\mathbb{R}})$ as the pre-dual of the quotient space $L^{\infty}(\mathbb{R})/\tilde{\mathbb{R}}$ where $\tilde{\mathbb{R}}$ is the set of all constant maps on $\mathbb{R}$. |
| title | Position of $L(X, Y)$ in $Lip_0(X, Y)$ |
| topic | Functional Analysis Primary 46B20, Secondary 46B03 |
| url | https://arxiv.org/abs/2506.09324 |