Common Hilbert Spaces
This module will give an overview of the most common Hilbert spaces and their basic properties.

### Common Hilbert Spaces

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:

• Standard inner product ${\mathbb{R}}^{n}$:
$\begin{array}{rcl}\hfill 〈x,y〉& \hfill =\hfill & {y}^{T}x\hfill \\ \hfill & \hfill =\hfill & \sum _{i=0}^{n-1}{x}_{i}{y}_{i}\hfill \end{array}$
• Standard inner product ${\mathbb{C}}^{n}$:
$\begin{array}{rcl}\hfill 〈x,y〉& \hfill =\hfill & \overline{{y}^{T}}x\hfill \\ \hfill & \hfill =\hfill & \sum _{i=0}^{n-1}{x}_{i}\overline{{y}_{i}}\hfill \end{array}$

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 $\left[a,b\right]$

Inner Product $〈f,g〉={\int }_{a}^{b}f\left(t\right)\overline{g\left(t\right)}dt$

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

Inner product $〈x,y〉=\sum _{i=-\infty }^{\infty }x\left[i\right]\overline{y\left[i\right]}$

Model for: discrete time, non-periodic signals

$f\in {L}^{2}\left(\mathbb{R}\right)$ is a finite energy function on all of $\mathbb{R}$.

Inner product $〈f,g〉={\int }_{-\infty }^{\infty }f\left(t\right)\overline{g\left(t\right)}dt$

Model for: continuous time, non-periodic signals

### Associated Fourier Analysis

Each of these 4 Hilbert spaces has a type of Fourier analysis associated with it.

• ${L}^{2}\left(\left[a,b\right]\right)$ → Fourier series
• ${\ell }^{2}\left(\left[0,n-1\right]\right)$ → Discrete Fourier Transform
• ${L}^{2}\left(\mathbb{R}\right)$ → Fourier Transform
• ${\ell }^{2}\left(\mathbb{Z}\right)$ → Discrete Time Fourier Transform

But all 4 of these are based on the same principles (Hilbert space).

Not all normed spaces are Hilbert spaces

For example: ${L}^{1}\left(ℝ\right)$, ${\parallel f\parallel }_{1}={\int }_{}^{}|f\left(t\right)|dt$. Try as you might, you can't find an inner product that induces this norm, i.e. a $〈·,·〉$ such that

$\begin{array}{rcl}\hfill 〈f,f〉& \hfill =\hfill & {\left({\int }_{}^{}{\left(|f\left(t\right)|\right)}^{2}dt\right)}^{2}\hfill \\ \hfill & \hfill =\hfill & {\left({\parallel f\parallel }_{1}\right)}^{2}\hfill \end{array}$

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.