Exercises 13 of Chapter 15 in Lax's Functional analysis (Sets of second category)

Question

Could anyone give some idea about the proof of the following theorem?

Answer 1

You can adapt the standard proof of the Open Mapping Theorem to obtain this result. (as found here: https://en.wikipedia.org/wiki/Open_mapping_theorem_%28functional_analysis%29)

Note that $M(X) = M \big (\bigcup_{k=1}^\infty kB_X(0,1) \big) = \bigcup_{k=1}^\infty M(kB_X(0,1))$. Since $M(X)$ is of second category, $kB_X(0,1)$ is not nowhere dense for some $k$.

Now, by mimicking the proof of the O.M.T., one can show that $M(B_X(0,1))$ contains an open ball about $0$ in $U$. It follows that $M$ is surjective.