Is there always a bounded linear functional on any vector space?

Question

Let $V$ be any non-zero vector space over the real numbers, not necessarily with a norm.

Does there always exist a linear map $\omega: V \to \mathbb{R}$ with $\omega \ne 0$ and such that

$$ \sup_{v \in V}|\omega(v)| < \infty$$

I guess this should somehow follow from Hahn-Banach but I'm not seeing what positive homogene subadditive form I can use here.

Answer 1

There is never any such map unless $V=\{0\}$. Proof: There exist $x$ such that $\omega (x) \neq 0$. Now look are $(\omega (nx))\equiv (n\omega (x))$. Obviously this sequence is not bounded.