Fixed point theorems in topological vector spaces and applications to random normed spaces

This chapter contains some results from the fixed point theory in topological vector spaces which are of special interest for the fixed point theory in random normed spaces. Namely, a random normed space (S, F, T) with a continuous t-norm T is a topological vector space which is not necessarily a locally convex space. It is known that a random normed space (S, F, T) is a locally convex space when T is a continuous t-norm of H-type. In the fixed point theory in a not necessarily locally convex topological vector spaces a very useful notion is that of an admissible subset which was introduced by Klee. Many important function spaces are admissible.