Measure, Integration & Real Analysis · Problem 1

Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · orthonormal-bases-verification

Problem

Verify that

\{e_k\}_{k \in \mathbf{Z}}

is an orthonormal family in

L^2\bigl((-\pi, \pi]\bigr)

, where for

k \in \mathbf{Z}

e_k: (-\pi, \pi] \to \mathbf{R}

is defined by

e_k(t) = \begin{cases} \dfrac{1}{\sqrt{\pi}} \sin(kt) & \text{if } k > 0, \\ \dfrac{1}{\sqrt{2\pi}} & \text{if } k = 0, \\ \dfrac{1}{\sqrt{\pi}} \cos(kt) & \text{if } k < 0. \end{cases}

(See Exercise 1 for useful formulas that will help with this verification.)