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Schwarz Lemma in infinite-dimensional spaces

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Abstract

We state and prove an extension of the Schwarz Lemma that involves infinite-dimensional spaces. Our generalized version contains some known variations of this classical result in geometric function theory. We derive our extension in the context of the Minkowski functional.

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Notes

  1. For a linear subspace \({\mathscr {X}}\subseteq X\), we will frequently speak of the \({\mathscr {X}}\)-section \(E \cap {\mathscr {X}}\) of E. In such situations, to say that the \({\mathscr {X}}\)-section \(E \cap {\mathscr {X}}\) is absorbing means that the \({\mathscr {X}}\)-section is absorbing relative to \({\mathscr {X}}\); that is, \(\mu _{E \cap {\mathscr {X}}} (x)\) is finite for any \(x \in {\mathscr {X}}\).

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  1. Manabu Ito

    Present address: , 10-20-101, Hirano-kita 1 chome, Hirano-ku, Osaka, 547-0041, Japan

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    1. Manabu Ito
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    Correspondence to Manabu Ito.

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    Communicated by Joachim Escher.

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    Ito, M. Schwarz Lemma in infinite-dimensional spaces. Monatsh Math 191, 735–748 (2020). https://doi.org/10.1007/s00605-020-01375-x

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    • DOI: https://doi.org/10.1007/s00605-020-01375-x

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