Let the vector space P^{2} have the inner product langle p,qrangle=int_{-1}^{1} p(x)q(x)dx. Find the following for p = 1 and q = x^{2}. (a) ⟨p,q⟩ (b) ∥p∥ (c) ∥q∥ (d) d(p,q)

Let the vector space ${P}^{2}$
have the inner product $⟨p,q⟩={\int }_{-1}^{1}p\left(x\right)q\left(x\right)dx.$
Find the following for
$\left(a\right)⟨p,q⟩\left(b\right)\parallel p\parallel \left(c\right)\parallel q\parallel \left(d\right)d\left(p,q\right)$
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l1koV

Given that the vector space P have the inner product
$⟨p,q⟩={\int }_{-1}^{1}p\left(x\right)q\left(x\right)dx.$
Also given that
We calculate the value of  as follows:
$⟨p,q⟩={\int }_{-1}^{1}p\left(x\right)q\left(x\right)dx$
$={\int }_{-1}^{1}{x}^{2}dx$
Therefore, the value of $⟨p,q⟩=\frac{2}{3}$