Complex extremal structure in spaces of continuous functions

[摘要] This paper considers the space Y = C(T, X) of all continuous and bounded functions from a topological space T to a complex normed space X with the sup 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 functions (mappings from T into the unit sphere of X). We prove that if T is 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 condition dim T < dim X-R is satisfied (where dim T is the covering dimension of T and X-R denotes X considered as a real normed space). If X is infinite-dimensional 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-dimensional complex normed space can be expressed as the average of three retractions of the unit ball onto the unit sphere. Really, such a representation is the best possible. (C) 1997 Academic Press.

[发布日期] 1997-07-15