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A Banach Space which is Fully 2-Rotund but not Locally Uniformly Rotund

Published online by Cambridge University Press:  20 November 2018

T. Polak  and
Brailey Sims
T. Polak
Affiliation:
Department of Mathematics, University of New EnglandArmidale, N.S.W. 2351., Australia
Brailey Sims
Affiliation:
Department of Mathematics, University of New EnglandArmidale, N.S.W. 2351., Australia
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Abstract

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A Banach space is fully 2-rotund if (xn) converges whenever ‖xn + xm‖ converges as m, n → ∞ and locally uniformly rotund if xn → x whenever ‖xn‖ and ‖(xn + x)/2‖ → ‖x‖.

We show that I2 with the equivalent norm

is fully 2-rotund but not locally uniformly rotund, thus answering in the negative a question first raised by Fan and Glicksberg in 1958.

Keywords

46B20
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 26 , Issue 1 , 01 March 1983 , pp. 118 - 120
Copyright
Copyright © Canadian Mathematical Society 1983

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