Scaled Sine Functions of Integer Multiples form Orthonormal Set

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

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

Then $S$ is an orthonormal set.

Proof
Consider the definite integral:
 * $I_{m n} = \displaystyle \int_0^l \map {\phi_m} x \, \map {\phi_n} x \rd x$

From Sine Function is Odd, each of $\map {\phi_n} x$ is an odd function.

From Odd Function Times Odd Function is Even, $\map {\phi_m} x \, \map {\phi_n} x$ is even.

That is:
 * $\paren {\sqrt {\dfrac 2 l} \sin \dfrac {m \pi x} l} \paren {\sqrt {\dfrac 2 l} \sin \dfrac {n \pi x} l}$

is an even function.

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

We have:

So:

Hence the result by definition of orthonormal set.