Vol. 38, No. 3, 1971

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Characterizations of uniform convexity

William Lee Bynum

Vol. 38 (1971), No. 3, 577–581

In this paper, three new characterizations of uniform convexity of a Banach space X are established. The characterization developed in Theorem 1 resembles the definition of the modulus of smoothness given by J. Lindenstrauss. The characterizations developed in Theorems 2 and 3 are interrelated, both involving the duality map of X into X. The methods used are adapted to give an abbreviated proof of a recent result of W. V. Petryshyn relating the strict convexity of X to the duality map of X into X.

Mathematical Subject Classification
Primary: 46B05
Received: 7 January 1971
Published: 1 September 1971
William Lee Bynum