Banach spaces, how to show if a subspace is Banach

Question

Is every subspace of a Banach space a Banach?
All I know is that… it is only yes iff it is closed and bounded and contains its limit points… any good point?

Answer 1

$W$ a subspace of a Banach space $X$ is Banach, i.e. complete, iff $W$ is closed.

Example of a subspace which is not closed:

Consider $X=\mathbb{R}$ as a vector space over $\mathbb{Q}$, complete with the usual norm. Let $W=\mathbb{Q}$, which is a subspace of $X$ but is not complete.