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Describes the extension of Hilbert spaces to infinite dimensions, including complete orthonormal sequences and Parseval's relation.

While up to now bases have been linked to finite-dimensional spaces and subspaces, it is possible to extend the concept to infinite-dimensional spaces.

Definition 1 Let ( X , R , + , · ) be a vector space. An infinite sequence of orthonormal vectors { e 1 , e 2 , ... } X is said to be a complete orthonormal sequence (CONS) in X if for every x X there exists a sequence α 1 , α 2 , . . . R such that x = i α i e i .

For the sake of concreteness, an infinite sum is defined as x = i = 1 α i e i = lim n i = 1 n α i e i . It is easy to see that α i = x , e i .

Lemma 1 An orthonormal sequence is complete if and only if the only vector in X orthogonal to each of the e i 's is the zero vector.

Example 1 For the space X of finite-energy complex-valued functions, R = C , a CONS is given by e k ( t ) = 1 2 π e j k t for k = 0 , ± 1 , ± 2 , . . . . These vectors are orthonormal:

e k , e l = 0 2 π 1 2 π e j k t e - j l t d t = 0 2 π 1 2 π e j ( k - l ) t d t = 1 k = l 0 k l

The coefficients are given by

c k = x , e k = 0 2 π x ( t ) e - j k t d t ,

and we obtain x = k c k e k . This is the sequence behind the Fourier series representation.

Example 2 Let X be the space of bandlimited functions x ( t ) (i.e., the set of functions with Fourier transform X ( f ) such that | X ( f ) | = 0 for all f [ - B , B ] ). A CONS for this space is given by

e k ( t ) = 1 2 B sinc 2 B t - k 2 B ,

where sinc ( t ) = ( sin ( π t ) ) / ( π t ) . It is possible to show that the functions are orthogonal to each other, i.e.,

e k , e l = δ k , l 1 k = l 0 k l .

If x is bandlimited, then it follows that x ( t ) = k c k e k ( t ) , with c k = x , e k = x ( k / ( 2 B ) ) . The preservation of the norm in the coefficients can also be extended from ONBs to CONS.

Theorem 1 (Completeness Relation) An orthonormal sequence e 1 , e 2 , . . . is complete for X if and only if the completeness relation holds for all x X :

x 2 = i | x , e i | 2 = i | c i | 2 .

The sequence { e i } is CONS if and only if

x = i = 1 x , e i e i = lim n i = 1 n x , e i e i = lim n x n ,

where x n = i = 1 n x , e i e i is the partial sum. We then have x 2 = x - x n 2 + x n 2 as these two components are orthogonal to each other. Applying a limit on both sides for n , we have

x 2 = lim n x - x n 2 + lim n x n 2 = 0 + lim n i = 1 n | x , e i | 2 = i = 1 | x , e i | 2 .

Theorem 2 Let X be a Hilbert space with a CONS { e 1 , e 2 , . . . } . Then for any x , y X , Parseval's relation holds: x , y = i x , e i y , e i ¯ .

Using the CONS, we can write the partial sums x n = i = 1 n x , e i e i and y n = i = 1 n y , e i e i . We then have

| x n , y n - x , y | = | x n , y - x n , y + x n , y n - x , y | = | x n , y n - y + x n - x , y | | x n , y n - y | + | x n - x , y | x n y n - y + x n - x y n

Letting n we have that the upper bound goes to zero, and therefore as n , we have x n , y n x , y . Therefore,

x , y = lim n x n , y n = lim n i = 1 n j = 1 n x , e i e i , y , e j e j = lim n i = 1 n j = 1 n x , e i y , e j ¯ e i , e j = lim n i = 1 n x , e i y , e i ¯ = i = 1 x , e i y , e i ¯ .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
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
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Signal theory. OpenStax CNX. Oct 18, 2013 Download for free at http://legacy.cnx.org/content/col11542/1.3
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