Below we will look at the four most common Hilbert spaces that you will have to deal with when discussing and manipulating signals and systems.
${\mathbb{R}}^{n}$ (reals scalars) and ${\mathbb{C}}^{n}$ (complex scalars), also called ${\ell}^{2}\left(\left[0,n-1\right]\right)$
$x=\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ \dots \\ {x}_{n-1}\end{array}\right)$ is a list of numbers (finite sequence). The inner product for our two spaces are as follows:
Model for: Discrete time signals on the interval $\left[0,n-1\right]$ or periodic (with period $n$) discrete time signals. $\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ \dots \\ {x}_{n-1}\end{array}\right)$
$f\in {L}^{2}\left(\left[a,b\right]\right)$ is a finite energy function on $\left[a,b\right]$
Model for: continuous time signals on the interval $\left[a,b\right]$ or periodic (with period $T=b-a$) continuous time signals
$x\in {\ell}^{2}\left(\mathbb{Z}\right)$ is an infinite sequence of numbers that's square-summable
Model for: discrete time, non-periodic signals
$f\in {L}^{2}\left(\mathbb{R}\right)$ is a finite energy function on all of $\mathbb{R}$.
Model for: continuous time, non-periodic signals
Each of these 4 Hilbert spaces has a type of Fourier analysis associated with it.
But all 4 of these are based on the same principles (Hilbert space).
For example: ${L}^{1}\left(\mathbb{R}\right)$, ${\parallel f\parallel}_{1}={\int}_{}^{}\left|f\left(t\right)\right|dt$. Try as you might, you can't find an inner product that induces this norm, i.e. a $\langle \xb7,\xb7\rangle $ such that
In fact, of all the ${L}^{p}\left(\mathbb{R}\right)$ spaces, ${L}^{2}\left(\mathbb{R}\right)$ is the only one that is a Hilbert space.
Hilbert spaces are by far the nicest. If you use or study orthonormal basis expansion then you will start to see why this is true.