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2005•Studia Scientiarum Mathematicarum HungaricaRequires access

Reflexivity, contraction functions and minimum-norm elements

Antonio Aizpuru, Francisco Javier García‐Pacheco

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Abstract

Here we present a new proof of Blatter's result: a normed space is complete if every bounded closed convex subset has an element of minimum norm. We also present geometrical conditions for the existence of minimum-norm elements in bounded closed convex sets. Also, we characterize reflexivity in the class of Banach spaces by means of contraction functions. Furthermore, we study what happens if we remove the completeness hypothesis.

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What this paper is about

Here we present a new proof of Blatter's result: a normed space is complete if every bounded closed convex subset has an element of minimum norm. We also present geometrical conditions for the existence of minimum-norm elements in bounded closed convex sets. Also, we characterize reflexivity in the class of Banach spaces by means of contraction functions. Furthermore, we study what happens if we remove the completeness hypothesis.

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Available abstract

Here we present a new proof of Blatter's result: a normed space is complete if every bounded closed convex subset has an element of minimum norm. We also present geometrical conditions for the existence of minimum-norm elements in bounded closed convex sets. Also, we characterize reflexivity in the class of Banach spaces by means of contraction functions. Furthermore, we study what happens if we remove the completeness hypothesis.

Key concepts: Mathematics, Norm (philosophy), Bounded function, Uniformly convex space, Normed vector space, Contraction (grammar), Regular polygon, Reflexive space

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