Scaled Sine Functions of Integer Multiples form Orthonormal Set

Theorem
For all $n \in \Z_{>0}$, let $\phi_n \left({x}\right)$ be the real function defined on the interval $\left({0, \,.\,.\, l}\right)$ as:
 * $\phi_n \left({x}\right) = \sqrt {\dfrac 2 l} \sin \dfrac {n \pi x} l$

Let $S$ be the set:
 * $S = \left\{ {\phi_n: n \in \Z_{>0} }\right\}$

Then $S$ is an orthonormal set.

Proof
Consider the definite integral:
 * $I_{m n} = \displaystyle \int_0^l \phi_m \left({x}\right) \phi_n \left({x}\right) \, \mathrm d x$

From Sine Function is Odd, each of $\phi_n \left({x}\right)$ is an odd function.

From Odd Function Times Odd Function is Even, $\phi_m \left({x}\right) \phi_n \left({x}\right)$ is even.

That is:
 * $\left({\sqrt {\dfrac 2 l} \sin \dfrac {m \pi x} l}\right) \left({\sqrt {\dfrac 2 l} \sin \dfrac {n \pi x} l}\right)$

is an even function.

Let $u = \dfrac {\pi x} l$.

We have:

So:

Hence the result by definition of orthonormal set.