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arXiv · 1907.01936

Emerging notions of norm attainment for Lipschitz maps between Banach spaces

Geunsu Choi·Yun Sung Choi·Miguel Martin
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Abstract

We classify several notions of norm attaining Lipschitz maps which were introduced previously, and present the relations among them in order to verify proper inclusions. We also analyze some results for the sets of Lipschitz maps satisfying each of these properties to be dense or not in $\Lip(X,Y)$. For instance, we characterize a Banach space $Y$ with the Radon-Nikodým property in terms of the denseness of norm attaining Lipschitz maps with values in $Y$. Further, we introduce a property called the local directional Bishop-Phelps-Bollobás property for Lipschitz compact maps, which extends the one studied previously for scalar-valued functions, and provide some new positive results.

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Geunsu Choi, Yun Sung Choi, Miguel Martin. 2019-07-03. Emerging notions of norm attainment for Lipschitz maps between Banach spaces. https://doi.org/10.1016/j.jmaa.2019.123600

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