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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

References

[A] Attouch, H., Variational convergence for functions and operators, Pitman, New York, 1984.Google Scholar
[B] Beauzamy, B., Introduction to Banach spaces and their geometry, North-Holland Mathematics Studies 68, North-Holland, 1985.Google Scholar
[BN] Borwein, J. M. and Noll, D., Second order differentiability of convex functions: A. D. Alexandrov 's Theorem in Hilbert space, preprint.Google Scholar
[Bou] Bourgain, R. D., Geometric aspects of convex sets with the Radon-Nikodym property, Lecture Notes in Mathematics 993, Springer- Verlag, 1983.Google Scholar
[C] Collier, J. B., The dual of a space with the Radon-Nikodym property, Pacific J. Math. 64(1976), 103–106.Google Scholar
[DGZ1] Deville, R., Godefroy, G. and Zizler, V., Smooth bump functions and geometry of Banach spaces, Mathematika, to appear.Google Scholar
[DGZ2] Deville, R., Renormings and smoothness in Banach spaces, Longman Monographs in Pure and Applied Mathematics, to appear.Google Scholar
[ET] Ekeland, I. and Temam, R., Convex Analysis and Variational Problems, Studies in Mathematics and its Applications, 1, North-Holland, 1976.Google Scholar
[En] Enflo, P., Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13 (1972), 281–288.Google Scholar
[F] Fabian, M., Lipschitz smooth points of convex functions and isomorphic characterizations of Hilbert spaces, Proc. London Math. Soc. 51(1985), 113–126.Google Scholar
[FWZ] Fabian, M., J. Whitfield, H. M. and Zizler, V., Norms with locally Lipschitzian derivatives, Israel J. Math. 44(1983),262–276.Google Scholar
[K] Kwapien, S., Isomorphic characterizations of inner product space by orthogonal series with vector valued coefficients, Studia Math. 44(1972), 583–595.Google Scholar
[LA-LI] J. Lasry, M. and Lions, P. L., A remark on regularization in Hilbert spaces, Israel J. Math. 55(1986), 257–266.Google Scholar
[PWZ] Pechanec, J., Whitfield, J. H. M. and Zizler, V., Norms locally dependent of finitely many coordinates, An. Acad. Brasil. Ciênc. 53(1981), 415–417.Google Scholar
[Vi ] Vanderwerff, J., Smooth approximations in Banach spaces, Proc. Amer. Math. Soc. 115(1992), 113–120.Google Scholar
[V2] Vanderwerff, J., Second-order Gâteaux differentiability and an isomorphic characterization of Hilbert spaces, Quart. J. Math. Oxford, to appear.Google Scholar