This problem has three questions (a), (b) and (c). I've done most of them, with a little conclusion in (c) undone, which I've thought about it for a long time.
(c) Let $f: X\to Y$ be a Borel measurable linear operator from a Banach space $X$ to a separable normed vector space $Y$, prove that $f$ is continuous. Hint: $B=\\{x\in X: ||f(x)||_Y<\frac{1}{2}\\}$ is a nonmeagre Borel set.
I wonder how to show that $B$ is nonmeagre. My method is trying to show $B$ is open, and then use Baire Category Theorem. But I'm unable to show $B$ contains a ball. How to solve this?
It's not a sophisticated riddle. $X=\bigcup_{n\in\Bbb{N}, n>0} nB$. A meager set is a countable union of nowhere-dense sets. A countable union of meager sets is a countable union of countable unions of nowhere-dense sets, and assuming the axiom of countable choice - it is meager too. So if $B$ were meager so would $X$.
Edit: You don't need the axiom of countable choice here. If $B$ is covered by a countable union of nowhere-dense sets - you can scale the same cover to get a cover for $nB$. Also, I neglected to mention it but you need to use the invariance of the topology to homothety of course.
Edit: I'm adding a photo of the exercise in case anyone is interested in what's discussed in the comments and can figure out why $Y$ needs to be separable.