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ON A WEAKLY UNIFORMLY ROTUND DUAL OF A BANACH SPACE

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  01 August 2012

J. R. GILES
J. R. GILES*
Affiliation:
School of Mathematical and Physical Sciences, The University of Newcastle, New South Wales 2308, Australia (email: [email protected])
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Abstract

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Every Banach space with separable second dual can be equivalently renormed to have weakly uniformly rotund dual. Under certain embedding conditions a Banach space with weakly uniformly rotund dual is reflexive.

Keywords

Gâteaux and uniformly Gâteaux differentiableweak and weak* uniform rotundityrenormingsreflexivity

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B10: Duality and reflexivity
Type
Retraction
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 1 , August 2015 , e1
Copyright
© 2012 Australian Mathematical Publishing Association Inc. 

References

Bollobás, B., ‘An extension of the theorem of Bishop and Phelps’, Bull. Lond. Math. Soc. 2 (1970), 181–182.CrossRefGoogle Scholar
Brown, A. L., ‘On the canonical projection of the third dual of a Banach space onto the first dual’, Bull. Aust. Math. Soc. 15 (1976), 351–354.CrossRefGoogle Scholar
Deville, R., Godefroy, G. and Zizler, V., Smoothness and Renorming in Banach Spaces (Longman, London, 1993).Google Scholar
Diestel, J., Geometry of Banach Spaces-Selected Topics, Lecture Notes in Mathematics, 485 (Springer, New York, 1975).CrossRefGoogle Scholar
Fabian, M., Habala, P., Hájek, P., Montesinos Santalucía, V., Pelant, J. and Zizler, V., Functional Analysis and Infinite Dimensional Geometry, Canad. Math. Soc., 8 (Springer, New York, 2001).CrossRefGoogle Scholar
Giles, J. R., ‘Uniformly weak differentiability of the norm and a condition of Vlasov’, J. Aust. Math. Soc. 21 (1976), 393–409.CrossRefGoogle Scholar
Giles, J. R., Gregory, D. A. and Sims, B., ‘Geometrical implications of upper semi-continuity of the duality on a Banach space’, Pacific J. Math. 79 (1978), 99–109.CrossRefGoogle Scholar
Hájek, P., ‘Dual renormings of Banach spaces’, Comm. Math. U. Carol. 37 (1996), 241–253.Google Scholar
Smith, M. A., ‘Rotundity and smoothness in conjugate spaces’, Proc. Amer. Math. Soc. 61 (1978), 232–234.CrossRefGoogle Scholar
Sullivan, F., ‘Geometrical properties determined by the higher duals of a Banach space’, Illinois. J. Math. 21 (1977), 315–331.CrossRefGoogle Scholar
Yorke, A. C., Smoothness and Rotundity in Banach Spaces, PhD Thesis, The University of Newcastle, NSW, 1977.CrossRefGoogle Scholar