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.
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