Vector Subspaces of the Set of Non-Norm-Attaining Functionals

García-Pacheco, Francisco Javier

doi:10.1017/s0004972708000348

Cited by 3 publications

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“…When is N A(X) spaceable? These questions have been deeply studied in [35] by Bandyopadhyay and Godefroy, who provided, among other results, conditions that ensure that N A(X) is not spaceable; see also the recent works [1,153] for more results on the linear structure of N A(X). Very recently, García-Pacheco and Puglisi [158] showed that every Banach space admitting an infinite dimensional separable quotient can be equivalently renormed in such a way that the set of its norm attaining functionals contains an infinite dimensional linear subspace.…”

Section: Some Remarks and Conclusion General Techniquesmentioning

confidence: 99%

Linear subsets of nonlinear sets in topological vector spaces

Bernal–González

Pellegrino

Seoane‐Sepúlveda

2013

Bull. Amer. Math. Soc.

197

149

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For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability. 109 5. Some remarks and conclusions. General techniques 113 About the authors 117 Acknowledgments 118 References 118Theorem 1.2 (Gurariy, 1966 [178]). The set of continuous nowhere differentiable functions on [0, 1] is lineable.Let us also recall that, although the set of everywhere differentiable functions in R is, in itself, an infinite dimensional vector space, in 1966 Gurariy obtained the following analogue of Theorem 1.1.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LINEAR SUBSETS OF NONLINEAR SETS IN TOPOLOGICAL VECTOR SPACES 73 Theorem 1.3 (Gurariy, 1966 [178]). The set of everywhere differentiable functions on [0, 1] is not spaceable in C[0, 1]. Also, there exist closed infinite dimensional subspaces of C[0, 1] all of whose members are differentiable on (0, 1).Somehow, what we are seeing is that what one could expect to be an isolated phenomenon can actually even have a nice algebraic structure (in the form of infinite dimensional subspaces). Unfortunately, and as we mentioned above, the Baire category theorem cannot be employed in the search for large subspaces such as the ones mentioned in the previous results. Let us now provide a more formal and complete definition for the above concepts and some other ones. Definition 1.4 (Lineability and spaceability [22,181,253]). Let X be a topological vector space and M a subset of X. Let μ be a cardinal number.(1) M is said to be μ-lineable (μ-spaceable) if M ∪ {0} contains a vector space (resp. a closed vector space) of dimension μ. At times, we shall refer to the set M as simply lineable or spaceable if the existing subspace is infinite dimensional.(2) We also let λ(M ) be the maximum cardinality (if it exists) of such a vector space. 1 (3) When the above linear space can be chosen to be dense in X, we shall say that M is μ-dense-lineable.Moreover, Bernal introduced in [69] the notion of maximal-lineable (and those of maximal-dense-lineable and maximal-spaceable), meaning that, when keeping the above notation, the dimension of the existing linear space equals dim(X). Besides asking for linear spaces, one could also study other structures, such as algebrability and some related ones, which were presented in [25,27,253]. Definition 1.5. Given a Banach algebra A, a subset B ⊂ A, and two cardinal numbers α and β, we say that:

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Section: Some Remarks and Conclusion General Techniquesmentioning

confidence: 99%

Linear subsets of nonlinear sets in topological vector spaces

Bernal–González

Pellegrino

Seoane‐Sepúlveda

2013

Bull. Amer. Math. Soc.

197

149

View full text Add to dashboard Cite

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Lineability of the set of supporting vectors

García-Pacheco

2021

RACSAM

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Norm-attaining functionals need not contain 2-dimensional subspaces

Rmoutil

2017

Journal of Functional Analysis

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G. Godefroy asked whether, on any Banach space, the set of normattaining functionals contains a 2-dimensional linear subspace. We prove that a recent construction due to C.J. Read provides an example of a space which does not have this property.This is done through a study of the relation between the following two sentences where X is a Banach space and Y is a closed subspace of finite codimension in X:We prove that these are equivalent if X is the Read's space.Moreover, we prove that any non-reflexive Banach space X with any given closed subspace Y of finite codimension at least 2 admits an equivalent norm such that implication (B) =⇒ (A) fails.

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Vector Subspaces of the Set of Non-Norm-Attaining Functionals

Abstract: An example is found of a nonreflexive Banach space X such that the union of {0} and the set X * \ NA(X ) of non-norm-attaining functionals on X contains no two-dimensional subspace.2000 Mathematics subject classification: 46B20, 46B03, 46B07.

Cited by 3 publications

References 14 publications

Linear subsets of nonlinear sets in topological vector spaces

Linear subsets of nonlinear sets in topological vector spaces

Lineability of the set of supporting vectors

Norm-attaining functionals need not contain 2-dimensional subspaces

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