Subalgebra of continuous functions - Imported Knowlpedia Content

A subalgebra of continuous functions on a topological space $X$ is a subset $A\subseteq C(X)$ (where $C(X)$ is the space of continuous functions) such that whenever $f,g\in A$ and $\alpha,\beta\in\mathbb{R}$ , the functions $\alpha f+\beta g$ and $f\cdot g$ (pointwise product) also belong to $A$ . If $A$ contains the constant function $1$ , it is called a unital subalgebra.

With pointwise operations, $C(X)$ is a commutative ring, and a subalgebra is a subset stable under the same structure. Subalgebras that separate points are central in the Stone–Weierstrass theorem.

Examples

On a closed interval $[a,b]$ , the set of restrictions of real polynomials to $[a,b]$ is a subalgebra of $C([a,b])$ .
On $[-1,1]$ , the set of even continuous functions $\{f\in C([-1,1]) : f(x)=f(-x)\}$ is a subalgebra of $C([-1,1])$ .