Let $V$ be a module of a Lie algebra $\mathfrak{g}$ and $V_{0}$ be the weight space of $V$ of weight $0$. $$ V_0 = \{ v\in V: h.v = 0, h \in \mathfrak{h} \}, $$ $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$.
It is said that $\dim V_0 = 1$. How to prove this? Thank you very much.
Edit: Let $V$ be a highest weight module of a Lie algebra $\mathfrak{g}$ with the highest weight $0$. Is it true that $\dim V = 1$? Thank you very much.