The Tychonoff theorem and invariant pseudodistances

Abstract

In this article we introduce a method of constructing functions with claimed properties by using the Tychonoff theorem. As an application of this method we show that the Carathéodory distance $c_D$ of convex domains $D$ in a complex, locally convex, Hausdorff, and infinite-dimensional topological vector space is approximated by the Carathéodory distances $c_{D\cap Y}$ in finite-dimensional linear subspaces $Y$. Originally this result is due to Dineen, Timoney, and Vigué who apply ultrafilters in their proof.