I am trying to solve this problem. Not sure if I am correct here

3. Consider the set S of real-valued polynomials P(x) of order n defined over the interval [-1, 1]: P(x) = Yar', re [-1, 1) and a; ER. (2) 1=0 Also define the notation: P(x) = x' for i = 0, ..., n. (a) Show that S is a vector space. (b) Show that {Po, P1, .... P.} is a basis of S. What is the dimension of S? (c) Given two polynomials P(x) and Q(x) in S, show that the following quantity is an inner product: (P,Q) =/ P(x)Q(x)dr. (3) (d) Is the basis defined in question (b) orthogonal with respect to this inner product? (Justify your answer) (e) Use the Gram-Schmidt orthogonalization process to construct the first four polynomials Po, P1, P2 and P in an orthonormal basis set starting from the basis of question (b). The resulting polynomials, called Legendre polynomials, play a very important role in applied mathematics. (f) Find the best approximation of the function f(x) = x - 2x + -x + 2 on span{Po}, span{ Po, Pi}, span { Po, PI, P2}, and span { Po, PI, P2, Pa}. In each case, determine the norm of the error. Plot f(r) and the various approximations on the same graph