In order to represent some signals, we will need some “atoms”. A convenient notion to define such atoms is the orthogonal bases.
Orthogonal basis A familly $\lbrace e_n\rbrace_{n\in\mathbb{N}}$ in an Hibert space $\mathcal{H}$ is said orthogonal iff for all $n\neq k$ $$ \langle e_k,e_n\rangle = 0\ . $$ In addition, if the family span $\mathcal{H}$, i.e. if for all $f\in \mathcal{H}$, there exists a sequence $\lambda_n$ of $\mathbb{C}$ such that $$ f = \sum_{n=0}^{+\infty} \lambda_n e_n $$ then $ \lbrace e_n\rbrace_{n\in\mathbb{N}} $ is an orthogonal basis of $\mathcal{H}$.
It will be convenient to suppose that the vectors $\lbrace e_n\rbrace$ are normalized, i.e. $\Vert e_n\Vert = 1$.
A direct usefull consequence is, if $\lbrace e_n\rbrace$ is an orthogonal basis of $\mathcal{H}$, we have for all $f\in \mathcal{H}$: $$ f = \sum_{n=0}^{+\infty} \frac{\langle f,e_n\rangle}{\Vert e_n\Vert } e_n\ . $$
Moreover, orthogonal basis allows one to preserve distance and angles.
Planchere-Parseval Let $\mathcal{H}$ be a Hilbert space and $\lbrace e_n\rbrace$ a normalized orthogonal basis of $\mathcal{H}$. Then, for all $f,g\in \mathcal{H}$, we have: $$ \langle f,g\rangle = \sum_{n=0}^{+\infty} \langle f,e_n\rangle \overline{\langle g,e_n\rangle}\ . $$ and $$ \Vert f\Vert ^2 = \sum_{n=0}^{+\infty} |\langle f,e_n\rangle|^2\ . $$