Inner product properties

Question

how prove the properties of inner produtcs on the formula

$$ \left( \sum_{j} a_{j}x^{j}, \sum_{i} b_{i}x^{i} \right) = \sum_{i,j} \frac{a_{j}b_{i}}{i+j+1} $$

to show that $(,)$ is a inner product on $R[x]$?

the properties are

$(x,y+z) = (x,y) + (x,z)$

$(x,y) = \overline{(y,x)}$

$(x,x) > 0$ if and only if $x \neq 0$

Answer 1

Hint: note that $$ (p(x),q(x))=\int_0^1 p(x)q(x)\,dx $$