Consider the inner product space P3(R) = span{1, x, x2, x3} with inner product = integral -1 to 1 p(x)q(x)

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Consider the inner product space P3(R) = span{1, x,
x2, x3} with inner product <p(x),
q(x)> = integral -1 to 1 p(x)q(x) dx, for p(x),
q(x) ∈ P3(R).
Let U be the subspace of P3(R) with basis β = ( 1 /√
2 , √(3/ 2) x, 1 /2 √(5/ 2) (1 − 3x2 ) )
.
(i) Show that β is an orthonormal basis for U.
(ii) Find the orthogonal projection of f(x) = 1 + x +
x3 onto U