Interior of a vector subspace

Let us denote $\Vert\cdot\Vert$ the norm on $E$.

Assume the interior of $F$ is not empty, which means that there exists a small open ball centered at an element of $F$, say $x_0$, of radius $\varepsilon$.
Now take $x\neq0$ in $E$. Hence $y:=x_0+x\cdot\frac{\varepsilon}{2\Vert x\Vert}$ is a vector such that $\Vert y-x_0\Vert =\frac{\varepsilon}{2}<\varepsilon$, hence it belongs to the ball formerly defined and so to $F$. But then $x$ is a linear combinaison of $y$ and $x_0$ so it belongs to $F$ since it is a vector space.

Finally, whether $F=E$ (and then the interior of $F$ is $E$ itself), whether the interior of $F$ is empty.