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2007•Bulletin of the Belgian Mathematical Society - Simon StevinOpen access

Functionals that do not attain their norm

Marı́a D. Acosta, A. Aizpuru, Richard M. Aron, Francisco Javier García‐Pacheco

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Abstract

We study the set of non-norm-attaining functionals on a Banach space, giving a sufficient condition for the density of this set. We also find a large class of Banach spaces for which the set of norm-attaining functionals is (dense-) lineable. In addition, among other results, we provide a new proof of the fact that every real Banach space can be equivalently renormed so that the set of non-norm-attaining functionals is non-dense.

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We study the set of non-norm-attaining functionals on a Banach space, giving a sufficient condition for the density of this set. We also find a large class of Banach spaces for which the set of norm-attaining functionals is (dense-) lineable. In addition, among other results, we provide a new proof of the fact that every real Banach space can be equivalently renormed so that the set of non-norm-attaining functionals is non-dense.

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Available abstract

We study the set of non-norm-attaining functionals on a Banach space, giving a sufficient condition for the density of this set. We also find a large class of Banach spaces for which the set of norm-attaining functionals is (dense-) lineable. In addition, among other results, we provide a new proof of the fact that every real Banach space can be equivalently renormed so that the set of non-norm-attaining functionals is non-dense.

Key concepts: Norm (philosophy), Banach space, Mathematics, Pure mathematics, Set (abstract data type), Banach manifold, Class (philosophy), Dual norm

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