Let $V$ be a complete normed space and $W$ is normed space. Let $A \subset V$ be closed subspace. If $ T:V \rightarrow W$ is bounded linear operator, then we need to prove that $T(A)$ is either first Baire Category or $T(A)=W$.
My attempt:
If $T$ is surjective then we are done. So, suppose that $T$ is not surjective, then I know we need to show that $T(A)$ is nowhere dense by showing it is contained in a closed set with empty interior but not sure how to prove this.
