15.6 Common hilbert spaces

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

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 $\left[a , b\right]$

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

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

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

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.