Analytic and $C^k$ approximations of norms in separable Banach spaces

Robert Deville; Vladimir Fonf; Petr Hájek

ArticleOriginal scientific text

Title

Analytic and $C^{k}$ approximations of norms in separable Banach spaces

Authors ¹, ², ³

Affiliations

Department of Mathematics, Université de Bordeaux, 351, Cours de la Libération, 33400 Talence, France
Department of Mathematics and Computer Sciences, Ben Gurion University of the Negev, Beer Sheva, Israel
Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Abstract

We prove that in separable Hilbert spaces, in

ℓ_{p} (ℕ)

for p an even integer, and in

L_{p} [0, 1]

for p an even integer, every equivalent norm can be approximated uniformly on bounded sets by analytic norms. In

ℓ_{p} (ℕ)

and in

L_{p} [0, 1]

for p ∉ ℕ (resp. for p an odd integer), every equivalent norm can be approximated uniformly on bounded sets by

C^{p}}

-smooth norms (resp. by

C^{p - 1}

-smooth norms).

Keywords

ENG

analytic norm, approximation, convex function, geometry of Banach spaces

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Title

Analytic and Ck approximations of norms in separable Banach spaces

Affiliations

Abstract

Keywords

Bibliography

Analytic and $C^{k}$ approximations of norms in separable Banach spaces