Let $0 \to V \to W \to L \to 0$ be a strict short exact sequence

of (complete) nuclear spaces, i.e. it is a short exact sequence of

(complete) nuclear spaces, all the maps are continuous, the map $V
\to W$ is a closed embeding, the topology on $V$ is induced from

$W$ and the map $W \to L $ is open. Let $U$ be a (complete) nuclear

space.
Is it true that the sequence obtained by completed tensor product with $U$
(i.e. $0 \to V \hat{\otimes} U \to W \hat{\otimes} U \to L
\hat{\otimes} U \to 0$)
is also strict short exact sequence?

We know that this is true if all the spaces are Frechet or if all

the spaces are dual Frechet, but is this true in general?