Some results on non-semireflexive spaces

In this paper we prove a result similar to a generalization due to M. Valdivia (4) of a theorem of V.L. Klee (1), in the context of real locally convex spaces which possess a metrizable locally convex topology coarser than the Mackey topology. In particular, we obtain some criteria of reflexivity for metrizable locally convex spaces. A particular case works for complex spaces.We thank Prof. Valdivia for his help and constant orientation This work follows from his suggestions.