User:RaisinBread/Sandbox

Work in progress
Stone's Theorem

Theorem
Let $K$ be a compact metric space and $\mathcal{C}(K,\R)$ be the Banach space of continuous functions from $K$ to $\R$.

Let $\mathcal{F}$ be a vector subspace of $\mathcal{C}(K,\R)$.

If $\mathcal{F}$ is such that:

1) $\forall f,g\in \mathcal{F}$, the functions $(f\lor g)(x)=$max$(f(x),g(x))$ and $(f\land g)(x)=$min$(f(x),g(x))$ are also in $\mathcal{F}$.

2) $\forall x,y\in K:x\neq y$, $\exists f\in\mathcal{F}:f(x)\neq f(y)$.

3) The unit function $f_1(x)=1$ is an element of $\mathcal{F}$.

Then, $\mathcal{F}$ is everywhere dense in $\mathcal{C}(K,\R)$