Linear Algebra Done Right · Problem 4

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · orthonormal-bases

Problem

Suppose

n

is a positive integer. Prove that

\frac{1}{\sqrt{2\pi}},\ \frac{\cos x}{\sqrt{\pi}},\ \frac{\cos 2x}{\sqrt{\pi}},\ \ldots,\ \frac{\cos nx}{\sqrt{\pi}},\ \frac{\sin x}{\sqrt{\pi}},\ \frac{\sin 2x}{\sqrt{\pi}},\ \ldots,\ \frac{\sin nx}{\sqrt{\pi}}

is an orthonormal list of vectors in

C[-\pi, \pi]

, the vector space of continuous real-valued functions on

[-\pi, \pi]

with inner product

\langle f, g \rangle = \int_{-\pi}^{\pi} fg.

Hint: The following formulas should help.

(\sin x)(\cos y) = \frac{\sin(x-y) + \sin(x+y)}{2}

(\sin x)(\sin y) = \frac{\cos(x-y) - \cos(x+y)}{2}

(\cos x)(\cos y) = \frac{\cos(x-y) + \cos(x+y)}{2}