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Extensions of uniformly smooth norms on Banach spaces

Published online by Cambridge University Press:  17 April 2009

R. Fry
Affiliation:
St. Francis Xavier University, Antigonish NS, Canada, e-mail: rfry@stfx.ca
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Abstract

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We give a characterisation for the extension of uniformly smooth norms from subspaces Y of superreflexive spaces X to uniformly smooth norms on all of X. This characterisation is applied to obtain results in various contexts.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Benyamini, Y. and Lindenstrauss, J., Volume 1, Geometric nonlinear functional analysis, American Mathematical Society Colloquium Publications 48 (American Mathematical Society, Providence R.I., 2000).Google Scholar
[2]Borwein, J.M., Fabian, M. and Vanderwerff, J., ‘Locally Lipschitz functions and bornological derivatives’, (preprint).Google Scholar
[3]Boiso, M. Cepedello, ‘Approximation of Lipschitz functions by Δ-convex functions in Banach spaces’, Israel. J. Math. 106 (1998), 269284.Google Scholar
[4]Deville, R., Godefroy, G. and Zizler, V., Smoothness and renorming in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics (Longman Scientific and Technical, Harlow, 1993).Google Scholar
[5]Fabian, M., ‘On extensions of norms from a subspace to the whole Banach space keeping their rotundity’, Studia Math. 112 (1995), 203211.Google Scholar
[6]Fabian, M., Zajícek, L. and Zizler, V., ‘On residuality of the set of rotund norms on a Banach space’, Math. Ann. 258 (1981/1982), 349351.CrossRefGoogle Scholar
[7]Fabian, M., Habala, P., Hájek, P., Santalucia, V.M., Pelant, J. and Zizler, V., Functional analysis and infinite-dimensional geometry, CMS Books in Mathematics 8 (Springer-Verlag, New York, 2001).Google Scholar
[8]Holmes, R.B., ‘Approximating best approximations’, Nieuw Arch. Wisk. (3) 14 (1966), 106113.Google Scholar
[9]John, K. and Zizler, V., ‘On extension of rotund norms’, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 24 (1976), 705707.Google Scholar
[10]Megginson, R.E., An Introduction to Banach space theory, Graduate Texts in Mathematics 183 (Springer-Verlag, New York, 1998).CrossRefGoogle Scholar
[11]McLaughlin, D., Poliquin, R., Vanderwerff, J. and Zizler, V., ‘Second order Gâteaux differentiable bump functions and approximations in Banach spaces’, Canad. J. Math. 45 (1993), 612625.Google Scholar
[12]Sundaresan, K. and Swaminathan, S., Geometry and nonlinear analysis in Banach spaces, Lectures Notes in Mathematics 1131 (Springer-Verlag, Berlin, 1985).Google Scholar
[13]Willard, S., General topology (Addison-Wesley Series in Mathematics, Reading MA, London, 1970).Google Scholar
[14]Zizler, V., ‘Smooth extensions of norms and complementability of subspaces’, Arch. Math. 53 (1989), 585589.CrossRefGoogle Scholar