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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

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit 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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