C(X) Spaces

Abstract

The space C(X) of all continuous complex- or real-valued functions on a topological space X plays an important role. This space, with pointwise multiplication, turns out to be an algebra. With a suitable topology, it is even a topological algebra. Thus the Stone-Weierstrass theorem can be formulated and proved. Furthermore, because of the lattice structure of the real line, there is a lattice structure on C(X). This is turn enables one to study some other, deeper properties of C(X). In particular, we prove the Banach-Stone theorem and other results which exploit the algebraic structure on C(X), which may not be available for C(X, 7), if X and Y are topological spaces and have no richer structure than that of the set of real numbers.