The Extension of Uniformly Continuous Banach Space-Valued Mappings

The authors investigated the uniform analogue of C-embedding, namely which subspaces S of a metric space M have the property that every uniformly continuous real-valued mapping on S can be extended to a uniformly continuous mapping on M. In this chapter, the authors consider the analogous question where the real line is replaced by the Banach space c₀ or l_∞. They introduce the basic notation and terminology. The authors summarize a number of conditions which are equivalent to the c₀ or l_∞-extension properties. They establish that every U-embedded subset of a normed linear space has the c₀ or l_∞-extension properties. The authors show that certain extension properties can be expressed in terms of the extension of uniformly continuous pseudometrics.