Make ℓ^2 (R) into an inner product space by defining ⟨x, y⟩ = ∫ 1 0 xy for all x, y in ℓ^2 (R).
(a) Apply the Gram-Schmidt procedure to the basis 1, x, x^2 to produce an orthonormal basis of â„“^2 (R).
(b) The differentiation operator (the operator that takes x to x') on â„“^2 (R) has an upper-triangular matrix with respect to the basis 1, x, x^2, which is not an orthonormal basis. Find the matrix of the differentiation operator on â„“^2 (R) with respect to the orthonormal basis produced in (a) and verify that this matrix is upper triangular, as expected from the proof of 6.37 from Done Right Linear Algebra 4th Edition by Axler.