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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.

n (reals scalars) and n (complex scalars), also called 2 0 n 1

x x 0 x 1 x n - 1 is a list of numbers (finite sequence). The inner product for our two spaces are as follows:

  • Standard inner product n :
    x y y x i 0 n 1 x i y i
  • Standard inner product n :
    x y y x i 0 n 1 x i y i

Model for: Discrete time signals on the interval 0 n 1 or periodic (with period n ) discrete time signals. x 0 x 1 x n - 1

f L 2 a b is a finite energy function on a b

Inner product

f g t a b f t g t
Model for: continuous time signals on the interval a b or periodic (with period T b a ) continuous time signals

x 2 is an infinite sequence of numbers that's square-summable

Inner product

x y i x i y i
Model for: discrete time, non-periodic signals

f L 2 is a finite energy function on all of .

Inner product

f g 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 a b → Fourier series
  • 2 0 n 1 → Discrete Fourier Transform
  • L 2 → Fourier Transform
  • 2 → 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 ( ) , 1 f t f t . Try as you might, you can't find an inner product thatinduces this norm, i.e. a · · such that
f f t f t 2 2 1 f 2
In fact, of all the L p spaces, L 2 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.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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