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

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.

• What is $F$ here? I think this wouldn't work for $F=\mathbb{C}$. – Peter Melech Jun 12 at 13:42
• All the polynomias such that deg(p) is equal or less than n – Juju9704 Jun 12 at 13:53
• Clear, the question is about the underlying field. If $F=\mathbb{C}$ You have to define the product differently $\langle p,q\rangle =\int_0^1 p(x)\overline{q(x)}dx$ to make this work. In both cases You just have to run through the definitions. You should really do that on Your own as an easy exercise – Peter Melech Jun 12 at 13:59