Banach Journal of Mathematical Analysis

Norm-attaining Lipschitz functionals

Vladimir Kadets, Miguel Martín, and Mariia Soloviova

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Abstract

We prove that for a given Banach space X, the subset of norm-attaining Lipschitz functionals in Lip0(X) is weakly dense but not strongly dense. Then we introduce a weaker concept of directional norm attainment and demonstrate that for a uniformly convex X the set of directionally norm-attaining Lipschitz functionals is strongly dense in Lip0(X) and, moreover, that an analogue of the Bishop–Phelps–Bollobás theorem is valid.

Article information

Source
Banach J. Math. Anal., Volume 10, Number 3 (2016), 621-637.

Dates
Received: 19 November 2015
Accepted: 28 November 2015
First available in Project Euclid: 22 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1471873728

Digital Object Identifier
doi:10.1215/17358787-3639646

Mathematical Reviews number (MathSciNet)
MR3541083

Zentralblatt MATH identifier
1358.46009

Subjects
Primary: 46B04: Isometric theory of Banach spaces
Secondary: 46B20: Geometry and structure of normed linear spaces 46B22: Radon-Nikodým, Kreĭn-Milman and related properties [See also 46G10] 47A30: Norms (inequalities, more than one norm, etc.)

Keywords
Bishop–Phelps–Bollobás theorem norm-attaining functional Lipschitz-free space Lipschitz functional uniformly convex Banach space

Citation

Kadets, Vladimir; Martín, Miguel; Soloviova, Mariia. Norm-attaining Lipschitz functionals. Banach J. Math. Anal. 10 (2016), no. 3, 621--637. doi:10.1215/17358787-3639646. https://projecteuclid.org/euclid.bjma/1471873728


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