TITLE: Non-Borel subspace of Banach space
QUESTION [7 upvotes]: Let $X$ be a separable Banach space, $M \subset X$ a linear subspace.  Must $M$ be a Borel set in $X$?
I believe the answer is "no," since I have seen authors who are careful to talk about "Borel subspaces".  But I have not been able to find a counterexample.
If the answer is indeed "no", does every infinite-dimensional separable Banach space contain a non-Borel dense linear subspace?

REPLY [3 votes]: Similar question was discussed before and answered (in particular to Bill's question in the comments) 
Are proper linear subspaces of Banach spaces always meager?