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This module introduces Hilbert spaces.

Now we consider inner product spaces with nice convergence properties that allow us to define countably-infinite orthonormalbases.

  • A Hilbert space is a complete inner product space. A complete
    The rational numbers provide an example of an incomplete set. We know thatit is possible to construct a sequence of rational numbers which approximate an irrational number arbitrarily closely. Itis easy to see that such a sequence will be Cauchy. However, the sequence will not converge to any rational number, and so the rationals cannot be complete.
    space is one where all Cauchy sequences converge to some vector within the space. For sequence x n to be Cauchy , the distance between its elements must eventually become arbitrarily small: ε ε 0 N ε n m n N ε m N ε x n x m ε For a sequence x n to be convergent to x , the distance between its elements and x must eventually become arbitrarily small: ε ε 0 N ε n n N ε x n x ε Examples are listed below (assuming the usual inner products):
    • V N
    • V N
    • V l 2 ( i.e. , square summable sequences)
    • V 2 ( i.e. , square integrable functions)
  • We will always deal with separable Hilbert spaces, which are those that have a countable
    A countable set is a set with at most a countably-infinite number of elements. Finite sets arecountable, as are any sets whose elements can be organized into an infinite list. Continuums ( e.g. , intervals of ) are uncountably infinite.
    orthonormal (ON) basis. A countable orthonormal basis for V is a countable orthonormal set S x k such that every vector in V can be represented as a linear combination of elements in S : y y V α k y k k α k x k Due to the orthonormality of S , the basis coefficients are given by α k x k y We can see this via: x k y x k n i 0 n α i x i n x k i 0 n α i x i n i 0 n α i x k x i α k where δ k i x k x i (where the second equality invokes the continuity of the inner product). In finite n -dimensional spaces ( e.g. , n or n ), any n -element ON set constitutes an ON basis. In infinite-dimensional spaces, wehave the following equivalences :
    • x 0 x 1 x 2 is an ON basis
    • If x i y 0 for all i , then y 0
    • y y V y 2 i i x i y 2 (Parseval's theorem)
    • Every y V is a limit of a sequence of vectors in span x 0 x 1 x 2
    Examples of countable ON bases for various Hilbert spaces include:
    • n : e 0 e N - 1 for e i 0 0 1 0 0 with "1" in the i th position
    • n : same as n
    • l 2 : δ i n i , for δ i n δ n i (all shifts of the Kronecker sequence)
    • 2 : to be constructed using wavelets ...
  • Say S is a subspace of Hilbert space V . The orthogonal complement of S in V , denoted S , is the subspace defined by the set x V x S . When S is closed, we can write V S S
  • The orthogonal projection of y onto S , where S is a closed subspace of V , is y ̂ i i x i y x i s.t. x i is an ON basis for S . Orthogonal projection yields the best approximation of y in S : y ̂ argmin x S y x The approximation error e y y ̂ obeys the orthogonality principle : e S We illustrate this concept using V 3 ( ) but stress that the same geometrical interpretation applies to any Hilbertspace.

A proof of the orthogonality principle is: e S i e x i 0 y y ̂ x i 0

y x i y ̂ x i j j x j y x j x i j j x j y x j x i j j y x j δ i j y x i

Questions & Answers

what is the stm
Brian Reply
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 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
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
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
how can I make nanorobot?
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.
Difference between extinct and extici spicies
Amanpreet Reply
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Leah Reply
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
Mapo Reply
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Mapo Reply
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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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