It's easy to check, isn't it?
etc.
Let P3 be the four-dimensional vector space of all polynomials of degree ≤ 3 with inner product
<f,g> = integral of f(x)g(x).dx from -1 to 1.
Apply Gram-Schmidt Process to basis {1, x, x2, x3} of P3 to obtain an orthonormal basis {z1, z2, z3, z4} for this inner product on P3.
I got the following but the numbers look extremely weird so I was wondering if others got the same answers:
z1 = 1/sqrt(2),
z2 = (3/sqrt(6))x,
z3 = (15/(2sqrt(10)))(x^2 - 1/3)
z4 = (35/(2sqrt(14)))(x^3 - (3/5)x)
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