Banach Journal of Mathematical Analysis

Norm-attaining Lipschitz functionals

Vladimir Kadets, Miguel Martín, and Mariia Soloviova

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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.)

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


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.

Export citation


  • [1] M. D. Acosta, R. M. Aron, D. García, and M. Maestre, The Bishop–Phelps–Bollobás theorem for operators, J. Funct. Anal. 254 (2008), no. 11, 2780–2799.
  • [2] M. Acosta, J. Becerra-Guerrero, D. García, and M. Maestre, The Bishop–Phelps–Bollobás theorem for bilinear forms, Trans. Amer. Math. Soc. 365 (2013), no. 11, 5911–5932.
  • [3] R. F. Arens and J. Eells Jr., On embedding uniform and topological spaces, Pacific J. Math. 6 (1956), 397–403.
  • [4] Y. Benyamini and J. Lindenstrauss. Geometric Nonlinear Functional Analysis, Vol. 1, Amer. Math. Soc. Colloq. Publ. 48, Amer. Math. Soc., Providence, 2000.
  • [5] E. Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc 67 (1961), no. 1, 97–98.
  • [6] B. Bollobás, An extension to the theorem of Bishop and Phelps, Bull. London Math. Soc. 2 (1970), 181–182.
  • [7] M. Chica, V. Kadets, M. Martín, S. Moreno-Pulido, and F. Rambla-Barreno, Bishop-Phelps-Bollobás moduli of a Banach space, J. Math. Anal. Appl. 412 (2014), no. 2, 697–719.
  • [8] J. Diestel, Geometry of Banach Spaces, Lecture Notes in Math. 485, Springer, Berlin, 1975.
  • [9] G. Godefroy, On norm attaining Lipschitz maps between Banach spaces, Pure Appl. Funct. Anal. 1 (2016), no. 1, 39–46.
  • [10] G. Godefroy and N. J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (2003), no. 1, 121–141.
  • [11] Y. Ivakhno, V. Kadets, and D. Werner, The Daugavet property for spaces of Lipschitz functions, Math. Scand. 101 (2007), no. 2, 261–279; Corrigendum in Math. Scand. 104 (2009), no. 2, 319.
  • [12] V. Kadets, Lipschitz mappings of metric spaces (in Russian), Isv. Vyssh. Uchebn. Zaved. Mat. (1985), no. 1, 30–34; English translation in Sov. Math. 29 (1985), no. 1, 36–41.
  • [13] M. Martín, J. Merí, and R. Payá, On the intrinsic and the spatial numerical range, J. Math. Anal. Appl. 318 (2006), no. 1, 175–189.
  • [14] C. Stegall, Optimization of functions on certain subsets of Banach spaces, Math. Ann. 236 (1978), no. 2, 171–176.
  • [15] N. Weaver, Lipschitz Algebras, World Scientific, River Edge, N.J., 1999.