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.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
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Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Damian
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LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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