Some stability properties for the
Bishop–Phelps–Bollobás property for Lipschitz maps

Chiclana, Rafael; Martı́n, Miguel

doi:10.4064/sm200427-7-4

Cited by 5 publications

(3 citation statements)

References 18 publications

(63 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Observe that, given a metric space M and a Banach space Y , then ASE(F(M ), Y ) is contained in SNA(M, Y ) since every element of ASE(F(M ), Y ) attains its norm at a strongly exposed point of F(M ), which is a molecule. As a consequence, we get the following result which extends previous results from [30] and [29]. We have not included the results which are covered by these two references.…”

supporting

confidence: 83%

Residuality in the set of norm attaining operators between Banach spaces

Jung¹,

Martín²,

Zoca³

2022

Preprint

View full text Add to dashboard Cite

We study the relationship between the residuality of the set of norm attaining functionals on a Banach space and the residuality and the denseness of the set of norm attaining operators between Banach spaces. Our first main result says that if C is a bounded subset of a Banach space X which admit an LUR renorming satisfying that, for every Banach space Y , the operators T from X to Y for which the supremum of T x with x ∈ C is attained are dense, then the G δ set of those functionals which strongly exposes C is dense in X * . This extends previous results by J. Bourgain and K.-S. Lau. The particular case in which C is the unit ball of X, in which we get that the norm of X * is Fréchet differentiable at a dense subset, improves a result by J. Lindenstrauss and we even present an example showing that Lindenstrauss' result was not optimal. In the reverse direction, we obtain results for the density of the G δ set of absolutely strongly exposing operators from X to Y by requiring that the set of strongly exposing functionals on X is dense and conditions on Y or Y * involving RNP and discreteness on the set of strongly exposed points of Y or Y * . These results include examples in which even the denseness of norm attaining operators was unknown. We also show that the residuality of the set of norm attaining operators implies the denseness of the set of absolutely strongly exposing operators provided the domain space and the dual of the range space are separable, extending a recent result for functionals. Finally, our results find important applications to the classical theory of norm-attaining operators, to the theory of norm-attaining bilinear forms, to the geometry of the preduals of spaces of Lipschitz functions, and to the theory of strongly norm-attaining Lipschitz maps. In particular, we solve a proposed open problem showing that the unique predual of the space of Lipschitz functions from the Euclidean unit circle fails to have Lindenstrauss property A.

show abstract

supporting

confidence: 83%

Residuality in the set of norm attaining operators between Banach spaces

Jung¹,

Martín²,

Zoca³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…There have been many efforts handling the heredity of norm attainment for operators and Lipschitz maps. We refer to [3,6] for these kinds of study which have been done recently, and we basically follow their ideas. Recall that an absolute norm |…”

Section: Denseness Of Norm Attaining Lipschitz Maps Toward Vectorsmentioning

confidence: 99%

Norm attaining Lipschitz maps toward vectors

Choi¹

2022

Preprint

View full text Add to dashboard Cite

We extend the recent result of G. Godefroy which concerns the existence of non-norm attaining Lipschitz maps in order to characterize the norm attainment toward vectors for Lipschitz maps in the general setting of underlying space. The main theorem of the present paper states that the existence of nonnorm attaining Lipschitz maps toward vectors on a large class of metric spaces is characterized by the finite-dimensionality of range space. As an extension of his counterexample, some denseness results on norm attaining Lipschitz maps toward vectors are also shown.Definition 1.1. A Lipschitz map f ∈ Lip 0 (M, Y ) is said to attain its norm toward y ∈ Y if there exists a sequence of distinct pairs {(p n , q n )} ⊆ M × M such that

show abstract

“…In the past few years, this topic has been intensively studied. We send the reader to [1,2,3,4,5,6,7,11,12,14,15,17,18,19] and the references therein.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

On isometric embeddings into the set of strongly norm-attaining Lipschitz functions

Dantas¹,

Medina²,

Quilis³

et al. 2022

Preprint

View full text Add to dashboard Cite

In this paper, we provide an infinite metric space M such that the set SNA(M ) of strongly norm-attaining Lipschitz functions does not contain a subspace which is isometric to c 0 . This answers a question posed by Antonio Avilés, Gonzalo Martínez Cervantes, Abraham Rueda Zoca, and Pedro Tradacete. On the other hand, we prove that SNA(M ) contains an isometric copy of c 0 whenever M is a metric space which is not uniformly discrete. In particular, the latter holds true for infinite compact metric spaces while it does not for proper metric spaces. Some positive results in the non-separable setting are also given.

show abstract

Some stability properties for the Bishop–Phelps–Bollobás property for Lipschitz maps

Cited by 5 publications

References 18 publications

Residuality in the set of norm attaining operators between Banach spaces

Residuality in the set of norm attaining operators between Banach spaces

Norm attaining Lipschitz maps toward vectors

On isometric embeddings into the set of strongly norm-attaining Lipschitz functions

Contact Info

Product

Resources

About