Function algebra

In functional analysis a Banach function algebra on a compact Hausdorff space X is unital subalgebra, A of the commutative C*-algebra C(X) of all continuous, complex valued functions from X, together with a norm on A which makes it a Banach 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 ${\displaystyle f(p)\neq f(q)}$.

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