A stochastic version of Fan's best approximation theorem

We establish some deterministic and random approximation results with the help of two continuous maps in the context of a metrizable topological vector space. These results generalize some well known results in approximation theory.

On the Dimension of a Complete Metrizable Topological Vector Space

The purpose of this note is to prove a result which is known to hold for Fréchet spaces [1, Chapitre II, §5, Exercise 24]. M. M. Day [2, p. 37] attributes the Banach space case to H. Löwig, although the earliest version that we have been able to find is that given by G. W. Mackey in [7, Theorem 1-1]. Recently H. E. Lacey has given an elegant proof for Banach spaces [5]. It is perhaps interesting to note that the non-locally convex case can be deduced from these known results which are established by duality arguments.