Spaces of Continuous Functions

This chapter is dedicated to the properties of spaces of continuous functions taking values in a semi-normed space. It presents the definitions of the space of continuous functions, the space of uniformly continuous functions, and variants of these spaces. The chapter compares these spaces and studies their completions and metrizability properties. It investigates the space of functions with compact support and studies continuous extensions, separation of variables, and sequential compactness in these spaces. A Fréchet space is a metrizable semi-normed space that is sequentially complete. A Banach space is a sequentially complete normed space. The chapter also recalls that an isomorphism from a separated semi-normed space to another is a continuous linear bijection whose inverse mapping is also linear and continuous.