What I have given is the Riesz representation theorem. I don't really understand how it works.
Let $\mathbb{P}_2$ be the vector space of polynomials with the highest degree of 2. Find the riesz representants for $1,x$ and $x^2$
$<p_1,p_2>:=\int_{-1}^{1}p_1(x)p_2(x)dx$
Riesz Theorem: Let $H$ be a Hilbert Space and $F \in H^{*}$. Then there exists exactly one $u \in H$ sucht that $<u,v>=F(v)$ for all $v$ in $H$
For example: Let u=x then <x,v> should equal F(v) for every v in $H$, how do I find the right F?
I didn't study functional analysis until now, but need to understand the results. I would be thankful if somebody could explain it to me.