I am aware that it may be a duplicate question, but I spent over an hour trying to find a proper proof and I failed. One may define inner product of polynomials $p, q$ as integral: $$\langle p, q \rangle = \int_a^b p(x)q(x) dx.$$ Inner product should satisfy following constraints:
- for each polynomials $p, q$: $\langle p, q \rangle = \langle q, p \rangle$ - this is quite obvious
- linearity - also easy to prove
- for each polynomial $p \neq 0$: $\langle p, p \rangle > 0$ - that's what troubles me
If I take any polynomial $p$, how can I be sure that the integral of $p^2$ is not $0$? I don't really see any argument that such a non-zero polynomial doesn't exist.