Abstract
We state and prove an infinitesimal version of the Schwarz Lemma that involves infinite-dimensional spaces. The emphasis here is on the term “infinitesimal.” 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
This striking result is well known; see Rudin [2]. It is straightforward to prove (without local-convexity assumption) that an originally bounded subset is weakly bounded but, to prove that a weakly bounded subset is originally bounded, the Hahn-Banach separation theorem, the Banach-Alaoglu theorem, and a generalization of the Banach-Steinhaus theorem are to be used.
References
Ito, M.: Schwarz lemma in infinite-dimensional spaces. Monatsh. Math. 191(4), 735–748 (2020)
Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc, New York (1991)
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It is my pleasure to thank an anonymous referee for the careful reading of the manuscript.
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Communicated by Adrian Constantin.
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Ito, M. Infinitesimal Schwarz Lemma in infinite-dimensional spaces. Monatsh Math 200, 119–130 (2023). https://doi.org/10.1007/s00605-022-01784-0
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DOI: https://doi.org/10.1007/s00605-022-01784-0