Norm vectorial space: an "enigma"

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.