A COUNTER EXAMPLE OF A HOMOMORPHISM THEOREM FOR LOCALLY CONVEX SPACES

Abstract

In this paper, we give a counter example of the following theorem given by F. Treves as Proposition 4.7 in [1]:
Let E, F be locally convex spaces and u: E→F be a continuous linear mapping. Then the following conditions are equivalent:
(a) u is a homomorphism;
(b) Im u_*=(u^-1'( {\\overline{0}}))⁰.
(a) implies (b). Conversely, if F is Hausdorff, (b) implies (a). But (b) does not necessarily imply (a). We give an example for which (b) is valid but (a) is not.