Let $T: X \to Y$ be a continuous linear transformation between the normed spaces $X$ and $Y$ with $\dim Y < \infty $. Let $P \subset X $ be a polyhedral convex set. Then my question is
Is $T(P)$ a closed set in $Y$ ?
Note that a polyhedral set has the form $P =\{ x \in X ~|~ \langle a^*_i \; , \; x \rangle \leq b_i ~~~ i =1,2,..,m \}.$
I know that if $X$ id finite-dimensional space then $T(P)$ is a polyhedral set so is closed. I'm not sure same result is true any normed spaces.