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A Generalization of Uniformly Rotund Banach Spaces

Published online by Cambridge University Press:  20 November 2018

Francis Sullivan
Francis Sullivan*
Affiliation:
The Catholic University of America, Washington, D.C
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Let X be a real Banach space. According to von Neumann's famous geometrical characterization X is a Hilbert space if and only if for all x, y ∈ X

Thus Hilbert space is distinguished among all real Banach spaces by a certain uniform behavior of the set of all two dimensional subspaces. A related characterization of real Lp spaces can be given in terms of uniform behavior of all two dimensional subspaces and a Boolean algebra of norm-1 projections [16]. For an arbitrary space X, one way of measuring the “uniformity” of the set of two dimensional subspaces is in terms of the real valued modulus of rotundity, i.e. for

The space is said to be uniformly rotund if for each 0 we have .

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 3 , 01 June 1979 , pp. 628 - 636
Copyright
Copyright © Canadian Mathematical Society 1979

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