> 
> On $P\_2$ define $\langle p(x),q(x)\rangle = p(1)q(1) + p(2)q(2)$ for
>  $p(x),q(x)$ in $P\_2$.
> 
> 
> I am very confused regarding how to tell that this is an inner
>  product.
> 
> 
> 

I know that in order to show its an inner product the following must hold true:

1) Symmetry: $\langle f, g \rangle = \langle g, f \rangle$

2) Linearity: $\langle \alpha f,g\rangle = \alpha \langle f,g\rangle$

3) Positive-definite: $\langle f, f \rangle \geq 0$. 

I have no idea how to prove it. How do I prove this is or is not an inner product?