Let $V$ be a normed vector space and $W$ be a proper closed subspace of $V$. I'm wondering if there is a continuous functional $\omega: V \to \mathbb{K}$ with $\Vert \omega \Vert = 1$ and $\omega(W) = \{0\}$.
I tried applying the Hahn-Banach theorem. It suffices to show that we can find a functional that is zero on $W$ and arbitrary norm, as we can always rescale.
Fix $v \in V \setminus W$. Define a functional on $W \oplus \mathbb{K}v$ by
$$w+ \lambda v \mapsto \lambda $$
but I can't find a subadditive function dominating this so I can't apply Hahn-Banach.
Best Answer
You can use the subadditive function $d(W,x)=\inf_{w\in W}\lVert x-w\rVert$ and consider $p(x)=\frac{d(W,x)}{d(W,v)}$. Notice for subadditivity that, for any two $w_1,w_2\in W$, $$\lVert x-w_1\rVert+\lVert y-w_2\rVert\ge \lVert x+y-w_1-w_2\rVert\ge d(W,x+y)$$ Hence, for all $w_2\in W$, $$\lVert y-w_2\rVert+d(W,x)\ge d(W,x+y)$$
Hence $d(W,x)+d(W,y)\ge d(W,x+y)$.