The minimal support for a continuous functional on a function space

Abstract:

Let ${C_p}(X)$ be the function space with the pointwise convergent topology over a Tychonoff space $X$ and $\xi$ a continuous real-valued function on ${C_p}(X)$. A closed subset $S$ of $X$ is called a support for $\xi$ if $\xi (f) = \xi (g)$ holds for any pair $(f,g)$ of elements of ${C_p}(X)$ such that ${f_{|S}} = {g_{|S}}$. It is proven that the minimal support for any real-valued continuous function on the space ${C_p}(X)$ exists.