Let Cpi pi f pi pi rightarrow R f is continuous denote the

Let C[-pi, pi] = {f: [-pi, pi] rightarrow R | f is continuous} denote the inner product space of continuous real-valued functions defined on the interval [-pi, pi] R, with inner product given by (f, g) = integral_-pi^pi f(x)g(x)dx, for every f, g epsilon C [-pi, pi]. Then, given any positive integer n epsilon Z_+, verify that the set of vectors {1/Squareroot 2 pi, sin(x)/Squareroot pi, sin(2x)/Squareroot pi, ... sin(nx)Squareroot pi, cos(x)Squareroot pi, cos(2x)Squareroot pi, ..., cos(nx)/Squareroot pi} is orthonormal. Let R_2[x] denote the inner product space of polynomials over R having degree at most two, with inner product given by (f, g) = integral_0^1 f(x)g(x) dx, for every f, g epsilon R_2[x]. Apply the Gram-Schmidt procedure to the standard basis {1, x, x^2} for R_2[x] in order to produce an orthonormal basis for R_2[x].

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