If Y is a subspace of any topological space X, then Y is always closed in itself (as you say, a space is both open and closed), but may be closed, open, both, or neither as a subset of X.
Let be E a Norm Vectorial space. And F Sub vectorial space of E. Thus, F can be considered as a vectorial space independently of E. And thus, as topological space, it is both open and closed. But we know that this is false. For example, the vectorial space of polynomial functions is not closed in the space of continous functions C[0,1] (since every continuous function is limit of a serie of polynomes - Weierstrass theroem). I found no way to find out the failure in the reasoning in de begining of this message. Many thanx for any help explaining where de faillure is.