TITLE: Open mapping theorem and second category
QUESTION [5 upvotes]: This seems like a fundamental result but I can not solve it of find an solution:
Let $M:X\rightarrow U$ be a bounded linear map between Banach spaces. Show that if the 
range of M is a set of second category of U; then the range is all of U.

REPLY [2 votes]: Assume that the range is of second category but the range of $M$ is not $U$. As $M(X)$ is not closed (otherwise $M(X)$ would be of first category, as it's a strict subspace), we can apply this result to $Y=\overline{M(X)}$.