Stone-Weierstrass: Literature

Question

Short question...

Does someone know some textbook, a paper or notes that treats:

An algebra of functions with identity that separates is dense within a function space.

Answer 1

You're probably looking for material related to the Stone Weierstrass Theorem. This is taken from Royden's Real Analysis:

Theorem [Stone-Weierstrass]: Let $X$ be a compact space and $A$ an algebra of continuous real-valued functions on $X$ which separates the points of $X$ and which contains the constant funtions. Then given any continuous real-valued function $f$ on $X$ and and $\epsilon > 0$ there is a function $g \in A$ such hat for all $x \in X$ we have $|g(x)-f(x)| < \epsilon$. In other words, $A$ is a dense subset of $C(X)$.

Royden approaches the subject through vector lattices of functions (operations $f \vee g$ and $f \wedge g$.) This approach is useful for defining abstract integrals. The Daniell Integral is defined as a positive linear functional $I$ on a lattice of real functions defined on some set $X$. The functional $I$ must satisfy $\lim_{n} I(\varphi_{n})=0$ for any sequence of positive real functions $\{ \varphi_{n} \}$ in the lattice which converges downward to $0$. The big theorem of the treatment is due to Marshall Stone, and generalizes a result of Daniell:

Theorem [Stone]: Let $L$ be a vector lattice of functions on $X$ with the property that if $f \in L$, then $1 \wedge f \in L$, and let $I$ be a Daniell integral on $I$. Then there is a $\sigma$-algebra $\mathcal{A}$ of subsets of $X$ and a measure $\mu$ on $\mathcal{A}$ such that each function $f$ on $X$ is integrable with respect to $I$ if and only if it is integrable with respect to $\mu$. Moreover, $$ I(f) = \int f d\mu. $$

Royden is a good source for anyone wanting to learn the subject on their own; Royden almost always makes the intuitive way of looking at the subject work in a clean, natural way.