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A Characterization of (Locally) Uniformly Convex Spaces in Terms of Relative Openness of Quotient Maps on the Unit Ball

✍ Scribed by J. Reif

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
148 KB
Volume
177
Category
Article
ISSN
0022-1236

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✦ Synopsis

Relative openness of quotient maps on the closed unit ball U of a normed linear space X is studied quantitatively. Particularly, it follows from the results that the quotient maps on X associated with the closed linear subspaces of X are equally relatively open on U if and only if X is locally uniformly convex. Also, X is locally uniformly convex if and only if for any family of linear maps defined on X, equal relative openness on X implies equal relative openness on U. Similarly, uniformly convex spaces can be characterized in terms of equal uniform relative openness of quotient maps on U.

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Let X be a complex strictly convex Banach space with an open unit ball B. For each compact, holomorphic and fixed-point-free mapping f: B Ä B there exists ! # B such that the sequence [ f n ] of iterates of f converges locally uniformly on B to the constant map taking the value !.