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Complex Extremal Structure in Spaces of Continuous Functions

✍ Scribed by A Jiménez-Vargas; J.F Mena-Jurado; J.C Navarro-Pascual

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
213 KB
Volume
211
Category
Article
ISSN
0022-247X

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✦ Synopsis

This paper considers the space Y s C T, X of all continuous and bounded functions from a topological space T to a complex normed space X with the sup Ž . norm. The extremal structure of the closed unit ball B Y in Y has been intensively studied when X is strictly convex, that is, in terms of its unitary Ž . functions mappings from T into the unit sphere of X . We prove that if T is Ž . completely regular and X has finite dimension, then every function in B Y is expressible as a convex combination of three unitary functions if and only if the

and X denotes X considered as a real normed space . If X is infinite-dimen-‫ޒ‬ sional the above decomposition is always possible without restrictions about T. These results are interesting when X is complex strictly convex. As a consequence we state a surprising fact: The identity function on the unit ball of an infinite-

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