A Pointwise Lipschitz Selection Theorem
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We prove that any correspondence (multi-function) mapping a metric space into a Banach space that satisfies a certain pointwise Lipschitz condition, always has a continuous selection that is pointwise Lipschitz on a dense set of its domain. We apply our selection theorem to demonstrate a slight improvement to a well-known version of the classical Bartle-Graves Theorem: Any continuous linear surjection between infinite dimensional Banach spaces has a positively homogeneous continuous right inverse that is pointwise Lipschitz on a dense meager set of its domain. An example devised by Aharoni and Lindenstrauss shows that our pointwise Lipschitz selection theorem is in some sense optimal: It is impossible to improve our pointwise Lipschitz selection theorem to one that yields a selection that is pointwise Lipschitz on the whole of its domain in general.
KeywordsSelection theorem Pointwise Lipschitz map Bartle-Graves Theorem
Mathematics Subject Classification (2010)54C65 54C60 (primary) 46B99 (secondary)
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The author would like to thank the MathOverflow community (Nate Eldredge in particular, for pointing out the example in Remark 3.5 to the author), and the anonymous referees of the paper for their constructive comments and suggestions.
The author’s research was funded by The Claude Leon Foundation.
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