Is a finite codimensional subspace always closed or dense?

Question

Let $E$ be a normed vector space. A standard result states that a hyperplane $H$ of $E$ is either closed or dense.
Does that remain true for finite codimensional subspaces ?
I tried proving that this is true using the following approach but I got stuck :
Let $H$ be a non closed finite codimensional subspace of $E$ and $F$ a complement of $H$. Let $(e_1,...,e_d)$ be a basis for $F$.
Let $x\in E$. Then $\exists h\in H,\ x_1,...,x_d\in \mathbb{R}$ such that $x=h+\sum_{i=1}^d x_ie_i.$
If $H$ is not closed in any of the subspaces $F_i=H\oplus\mathbb{R}e_i$ then $H$ is dense in $F_i$ $\forall i$ so $x_ie_i$ is a limit of a sequence $(h_{i,n})$ of elements of $H$ $\forall i$. And so $x=\lim_{n\to\infty} (h+\sum_{i=1}^{d}h_{i,n})$. So $H$ is dense in E.
But what if $H$ is actually closed in one of the subspaces $F_i$ ? (Is that possible even though we supposed $H$ is not closed in $E$ ?)
Thank you in advance for your help.

Answer 1

It is not true for codimension $2$. If $H$ is a dense hyperplane in $E$, and $E$ is a closed hyperplane in $F$, then $H$ is a subspace of codimension $2$ in $F$ which is neither dense nor closed.