Can a proper subspace of a Banach space be a countable intersection of open sets?

Question

Let X be a Banach space and suppose E is a dense linear subspace which is a Gδ -set. Show that E = X.

By the Baire Category theorem we can show that E is non-meager (second category), but I don't know where to continue from here. If I could show that every proper subspace is meager, then I would be done, but I don't know if this is the correct approach.