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2013•Revista de la Unión Matemática ArgentinaOpen access

Translations, norm-attaining functionals, and elements of minimum norm

Francisco Javier García‐Pacheco

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Abstract

In this paper we continue a work that James started in 1971 about norm-attaining functionals on non-complete normed spaces by proving that every functional on a normed space is norm-attaining if and only if every proper, closed, convex subset with non-empty interior can be translated to have a non-zero, minimum-norm element. We also study this type of spaces when they are non-complete. Finally, we consider translations and elements of maximum norm.

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What this paper is about

In this paper we continue a work that James started in 1971 about norm-attaining functionals on non-complete normed spaces by proving that every functional on a normed space is norm-attaining if and only if every proper, closed, convex subset with non-empty interior can be translated to have a non-zero, minimum-norm element. We also study this type of spaces when they are non-complete. Finally, we consider translations and elements of maximum norm.

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Available abstract

In this paper we continue a work that James started in 1971 about norm-attaining functionals on non-complete normed spaces by proving that every functional on a normed space is norm-attaining if and only if every proper, closed, convex subset with non-empty interior can be translated to have a non-zero, minimum-norm element. We also study this type of spaces when they are non-complete. Finally, we consider translations and elements of maximum norm.

Key concepts: Norm (philosophy), Mathematics, Dual norm, Normed vector space, Regular polygon, Pure mathematics, Geometry, Epistemology

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