Injectivity of the scalar product

I want to show that $\displaystyle \hat{A}: V^* \times V \rightarrow \mathbb{R}$ (where $\displaystyle V$ is a vector space and $\displaystyle V^*$ is it's dual space) defined by:

$\displaystyle \hat{A}(v^*,v) \equiv <v^*, Av>$

is injective.

I have a proof from a book, but I don't quite understand what it's doing. It starts by:

Quote:

Let $\displaystyle \hat{A}=0$. Then $\displaystyle <v^*,Av>=0 \ \forall v^* \in V^*, \ v \in V$.

Now since the scalar product is definite, this implies $\displaystyle Av=0 \ \forall v \in V$ and thus $\displaystyle A=0$.

Haven't they just proven it for just 0?

Also, what does "scalar product is definite" mean?