Linear subspace of Banach space containing unit ball

Question

Am I right that any linear subspace of Banach space which contain unit ball is whole space?

Answer 1

Yes, take any vector in the space and divide it by (a little more than) its and you'd end up with a vector within the unit ball. Hence, every vector in a Banach space is conlinear to a vector in the unit ball and so the unit ball spans the entire space.

Answer 2

You have that for every $ε>0 $ $ B(0,ε)=B(x,ε)-x$ and because $B(0,λε)=λB(0,ε)$ for every $λ>0$ (linear subspace) we have that $X=\cup_{λ>0} B(0,λε)$