More smoothly real compact spaces

Abstract:

A topological space $X$ is called $\mathcal {A}$-real compact if every algebra homomorphism from $\mathcal {A}$ to the reals is an evaluation at some point of $X$, where $\mathcal {A}$ is an algebra of continuous functions. Our main interest lies on algebras of smooth functions. Arias-de-Reyna has shown that any separable Banach space is smoothly real compact. Here we generalize this result to a huge class of locally convex spaces including arbitrary products of separable Fréchet spaces.