Complex Extremal Structure in Spaces of Continuous Functions

This paper considers the space YsC 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 is satisfied where dim T is the covering dimension of T R and X denotes X considered as a real normed space.. If X is infinite-dimen- R 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-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