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
$\left[a,b\right]$
${\ell}^{2}\left(\mathbb{Z}\right)$ → Discrete Time Fourier Transform
But all 4 of these are based on the same principles (Hilbert space).
Important note:
Not all normed spaces are Hilbert
spaces
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
$\u27e8\xb7,\xb7\u27e9$ such that