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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

I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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
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How can I make nanorobot?
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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