Orthogonal Functions and Orthogonal Sets
Definition
Inner Product
For two functions $f$, $g$ defined on the interval $[a,b]$, the inner product is defined as follows.
$$ \braket{f , g} := \int_{a}^b f(x) g(x) dx $$
If $f, g$ are complex functions,
$$ \braket{f, g} := \int_{a}^{b} f(x) \overline{g(x)} dx $$
Here $\overline{z}$ is the complex conjugate of $z$.
Orthogonal Functions
If two complex functions $f$, $g$ satisfy the equation below, we say that $f$, $g$ are orthogonal on the interval $[a,b]$.
$$ \braket{f, g} = \int_{a}^{b} f(x) \overline{g(x)} dx = 0 $$
Since the inner product of two functions was defined above as an integral, it is natural to say they are orthogonal when the integral value is $0$.
Orthogonal Set
If functions $\phi_{1}$, $\phi_2$, $\phi_{3}$, $\dots$ satisfy the following equation, the set $\left\{\phi_{i}\right\}$ of these functions is called an orthogonal set.
$$ \braket{\phi_{m}, \phi_{n}} = \int_{a}^b \phi_{m} (x) \overline{ \phi_{n}(x) } dx = 0 \quad (m\ne n) $$
Normalization
The norm of a function $f$ is defined as follows.
$$ \left\| f \right\| := \sqrt{\braket{f, f}} = \left( \int_{a}^b \left| f(x) \right| ^2 dx \right) ^{ \frac{1}{2} } $$
For an arbitrary function $f$, multiplying by an appropriate constant so that the norm of $f$ becomes $1$ is called normalize. The function $f_{\text{normal}}$ obtained by normalizing the function $f$ is
$$ f_{\mathrm{normal}} = \frac{1}{ \left\| f \right\| }f $$
Orthonormal Set
If every element of an orthogonal set $\left\{ \phi_{1}, \phi_{2}, \cdots \right\}$ is a normalized function, the set is called an orthonormal set. That is, if the following holds for all $n, m$, it is an orthonormal set.
$$ \braket{\phi_{m}, \phi_{n}} = \int_{a}^b \phi_{m} (x) \overline{ \phi_{n}(x) } dx=\delta_{mn} $$
Here $\delta_{mn}$ is the Kronecker delta.
Explanation
Orthogonality of functions is a concept that naturally generalizes to the space of functions the statement that two vectors in Euclidean space are orthogonal when their inner product is $0$. Since the inner product defined above is precisely the inner product of the $L^{2}$ space, one can treat square-integrable functions like vectors and discuss their orthogonality. A representative application is the Fourier series; since the set of trigonometric functions forms an orthogonal set on the interval $[-\pi, \pi]$, when expanding a function into a series of trigonometric functions, each coefficient can be simply found with a single inner product.