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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:
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 $\mathbb{R}$ .
Each of these 4 Hilbert spaces has a type of Fourier analysis associated with it.
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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