Let $X$ be a Banach space, $Y$ a normed vector space, and $T: X \rightarrow Y$ a bounded linear map. Assume that $T(X)$ is of second category in $Y$. Show that $T$ is surjective and $T(\mathcal{U})$ is open whenever $\mathcal{U}$ is open. Show that, if $T$ is also injective, then $T^{-1}$ exists and is bounded.

Give an example of a continuous map $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(\mathbb{R})$ is of second category in $\mathbb{R}$ but $f$ is not surjective. Give an example of a continuous surjective map $f: \mathbb{R} \rightarrow \mathbb{R}$ which does not take open sets to open sets.