# 15.6 Common hilbert spaces

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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 manipulatingsignals and systems.

$\mathbb{R}^{n}$ (reals scalars) and $\mathbb{C}^{n}$ (complex scalars), also called ${\ell }^{2}(\left[0 , n-1\right])$

$x=\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ \dots \\ {x}_{n-1}\end{array}\right)$ is a list of numbers (finite sequence). The inner product for our two spaces are as follows:

• Standard inner product $\mathbb{R}^{n}$ :
$x\dot y=y^Tx=\sum_{i=0}^{n-1} {x}_{i}{y}_{i}$
• Standard inner product $\mathbb{C}^{n}$ :
$x\dot y=\overline{y^T}x=\sum_{i=0}^{n-1} {x}_{i}\overline{{y}_{i}}$

Model for: Discrete time signals on the interval $\left[0 , n-1\right]$ or periodic (with period $n$ ) discrete time signals. $\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ \dots \\ {x}_{n-1}\end{array}\right)$

$f\in {L}^{2}(\left[a , b\right]())$ is a finite energy function on $\left[a , b\right]$

## Inner product

$f\dot g=\int_{a}^{b} f(t)\overline{g(t)}\,d t$
Model for: continuous time signals on the interval $\left[a , b\right]$ or periodic (with period $T=b-a$ ) continuous time signals

$x\in {\ell }^{2}(\mathbb{Z})$ is an infinite sequence of numbers that's square-summable

## Inner product

$x\dot y=\sum_{i=()}$ x i y i
Model for: discrete time, non-periodic signals

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

## Inner product

$f\dot g=\int_{()} \,d t$ f t g t
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.

• ${L}^{2}(\left[a , b\right])$ → Fourier series
• ${\ell }^{2}(\left[0 , n-1\right]())$ → Discrete Fourier Transform
• ${L}^{2}(\mathbb{R})$ → Fourier Transform
• ${\ell }^{2}(\mathbb{Z})$ → 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: ${L}^{1}\left(ℝ\right)$ , $(, f)=\int \left|f(t)\right|\,d t$ . Try as you might, you can't find an inner product thatinduces this norm, i.e. a $·\dot ·$ such that
$f\dot f=\int \left|f(t)\right|^{2}\,d t^{2}=(, f)^{2}$
In fact, of all the ${L}^{p}(\mathbb{R})$ spaces, ${L}^{2}(\mathbb{R})$ 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.

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