$V=${$p(x)=a_0+...+a_nx^n:a_j\in F$} and $<p*q>=\int_0^1p(x)q(x)dx$ over V. Prove that V is a vector space of inner product such that $\dim V=n+1$

Question

If $V=F_n[x]=\{p(x)=a_0+...+a_nx^n:a_j\in F\}$ ($\deg(p)$ is no less than n) and consider the inner product

$$\langle p*q\rangle=\int_0^1p(x)q(x)\,dx$$

over V

Prove that V is a vector space with inner product such that $\dim V=n+1$

I already did some other questions but I cannot solve this one, I'd be thankful to get some help.