So here is my problem,
I am trying to show that,
Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded map. Then $T(X)$ is closed if $Y/T(X)$ is of finite dimension.
While thinking about the given setting I found the following,
Let $\pi:Y\rightarrow Y/T(X)$ be the canonical projection into the quotient. Then since $\{\overline{0}\}\in Y/T(X)$ is closed it will follw that $\pi^{-1}(\{\overline{0}\})=T(X)$ is closed by continuity of $\pi$. I am sure that i made a fundamental mistake since I didnt use almost any of the given informations. But I am not sure wehre the problem is... Is it the continuity of $\pi$ or is there some case where $\{\overline{0}\}$ is not closed?
I hoped someone could help me to solve my confusion. Thank you!!