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Second-Order Gâteaux Differentiable Bump Functions and Approximations in Banach Spaces

Published online by Cambridge University Press:  20 November 2018

D. McLaughlin ,
R. Poliquin ,
J. Vanderwerff  and
V. Zizler
D. McLaughlin
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, T6G 2G1
R. Poliquin
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, T6G 2G1
J. Vanderwerff
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, T6G 2G1
V. Zizler
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, T6G 2G1
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Abstract

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In this paper we study approximations of convex functions by twice Gâteaux differentiate convex functions. We prove that convex functions (respectively norms) can be approximated by twice Gâteaux differentiate convex functions (respectively norms) in separable Banach spaces which have the Radon-Nikody m property and admit twice Gâteaux differentiable bump functions. New characterizations of spaces isomorphic to Hilbert spaces are shown. Locally uniformly rotund norms that are limits of Ck-smooth norms are constructed in separable spaces which admit Ck-smooth norms.

Keywords

46B2046B2246C05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 3 , 01 June 1993 , pp. 612 - 625
Copyright
Copyright © Canadian Mathematical Society 1993

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