Continuous functions on a compact Hausdorff space

In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space ${\displaystyle X}$ with values in the real or complex numbers. This space, denoted by ${\displaystyle {\mathcal {C}}(X),}$ is a vector space with respect to the pointwise addition of functions and scalar multiplication by constants. It is, moreover, a normed space with norm defined by

the uniform norm. The uniform norm defines the topology of uniform convergence of functions on ${\displaystyle X.}$ The space ${\displaystyle {\mathcal {C}}(X)}$ is a Banach algebra with respect to this norm.(Rudin 1973, §11.3)