Banach function algebra

A function algebra is said to vanish at a point p if f(p) = 0 for all ${\displaystyle (f\in A)}$. A function algebra separates points if for each distinct pair of points ${\displaystyle (p,q\in X)}$, there is a function ${\displaystyle (f\in A)}$ such that ${\displaystyle f(p)\neq f(q)}$.
If the norm on ${\displaystyle A}$ is the uniform norm (or sup-norm) on ${\displaystyle X}$, then ${\displaystyle A}$ is called a uniform algebra. Uniform algebras are an important special case of Banach function algebras.