$X$ and $Y$ is topological vector spaces.
$T$ is linear map between this two spaces. Also given that $ker(T)$ is closed and $Y$ is finite dimensional. We have to show that $T$ is continuous and open map.
I have already shown that $T$ is continuous. Without using the fact that $Y$ is finite dimensional. If I able to show $T$ is continuous linear map from that we can conclude that $T$ is bounded in some neighborhood $V$ of zero. From that we can say $T$ is open map. That is there is an open set contains in the $T(X)$.
I don't understand where we required the fact that $Y$ is finite dimensional?