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.

  • $\begingroup$ What is $F$ here? I think this wouldn't work for $F=\mathbb{C}$. $\endgroup$ – Peter Melech Jun 12 at 13:42
  • $\begingroup$ All the polynomias such that deg(p) is equal or less than n $\endgroup$ – Juju9704 Jun 12 at 13:53
  • 1
    $\begingroup$ 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 $\endgroup$ – Peter Melech Jun 12 at 13:59

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