On simultaneous extension of continuous functions
H. H. Corson, Joram Lindenstrauss · Bulletin of the American Mathematical Society · 1965
Let S be a compact Hausdorff space and let K be a closed subset of S. Denote by C(S) (respectively, C(K)) the Banach space of all continuous real-valued functions on 5 (respectively, K) with the supremum norm.A bounded linear operator T from C(K) to C(S) is called a simultaneous extension (s.e.) operator if the restriction of Tf to K is equal to ƒ for every ƒ £ C(K).Put V (K, S) = inf{||r||; T is an s.e.operator from K to S}. (rj(K, S) = oo if there exists no s.e.operator from K to S.) Several authors (for example, Borsuk, Kakutani, Dugundji and Arens, cf. the expository paper [4] for references) have considered this notion of simultaneous extension of continuous functions.It is known that, if K is metrizable, then rj(K, S) = 1 for every SZ)K (cf. the recent paper [3] for a much stronger result), and examples of K and 5 for which t](K, S) = 00 are known.As far as we know, in all examples considered thus far either one of these two extreme situations occurred.In this note we find all the possible values of rj(K, S) for K the onepoint compactification of an uncountable set (which is, in a sense, the simplest nonmetrizable compact Hausdorff space).The result we obtain is somewhat surprising and it indicates that the study of the behaviour of rj(K, S) for more general K may be of interest.We intend to consider this question as well as the more general question of extending maps into nonmetrizable compact convex sets in a future paper (cf.also Proposition 1 in this note).