Given that the vector space P have the inner product

\(\langle p,q\rangle=\int_{-1}^{1} p(x)q(x)dx.\)

Also given that \(p = 1\ and\ g = x^{2}\)

We calculate the value of \(\) as follows:

\(⟨p, q⟩ =\int_{-1}^{1} p(x)q(x)dx\)

\(= \int_{-1}^{1} x^{2} dx\)

Therefore, the value of \(\langle p,q\rangle=\frac{2}{3}\)