# 6.4 Hilbert spaces

 Page 1 / 1
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: $\forall \epsilon , \epsilon > 0\colon \exists {N}_{\epsilon }\colon \forall n, m, (n\ge {N}_{\epsilon })\land (m\ge {N}_{\epsilon })\colon ({x}_{n}-{x}_{m})< \epsilon$ For a sequence $\{{x}_{n}\}$ to be convergent to x , the distance between its elements and $x$ must eventually become arbitrarily small: $\forall \epsilon , \epsilon > 0\colon \exists {N}_{\epsilon }\colon \forall n, n\ge {N}_{\epsilon }\colon ({x}_{n}-x)< \epsilon$ Examples are listed below (assuming the usual inner products):
• $V=\mathbb{R}^{N}$
• $V=\mathbb{C}^{N}$
• $V={l}_{2}$ ( i.e. , square summable sequences)
• $V={ℒ}_{2}$ ( 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 $\mathbb{R}$ ) 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 $S$ : $\forall y, y\in V\colon \exists \{{\alpha }_{k}\}\colon y=\sum_{k} {\alpha }_{k}{x}_{k}$ Due to the orthonormality of $S$ , the basis coefficients are given by ${\alpha }_{k}={x}_{k}\dot y$ We can see this via: ${x}_{k}\dot y={x}_{k}\dot \lim_{n\to }n\to$ 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 $\delta (k-i)={x}_{k}\dot {x}_{i}$ (where the second equality invokes the continuity of the inner product). In finite $n$ -dimensional spaces ( e.g. , $\mathbb{R}^{n}$ or $\mathbb{C}^{n}$ ), any $n$ -element ON set constitutes an ON basis. In infinite-dimensional spaces, wehave the following equivalences :
• $\{{x}_{0}, {x}_{1}, {x}_{2}, \dots \}$ is an ON basis
• If ${x}_{i}\dot y=0$ for all $i$ , then $y=0$
• $\forall y, y\in V\colon (y)^{2}=\sum_{i} \left|{x}_{i}\dot y\right|^{2}$ (Parseval's theorem)
• Every $y\in V$ is a limit of a sequence of vectors in $\mathrm{span}(\{{x}_{0}, {x}_{1}, {x}_{2}, \dots \})$
Examples of countable ON bases for various Hilbert spaces include:
• $\mathbb{R}^{n}$ : $\{{e}_{0}, \dots , {e}_{N-1}\}$ for ${e}_{i}=\begin{pmatrix}0 & \dots & 0 & 1 & 0 & \dots & 0\\ \end{pmatrix}^T$ with "1" in the ${i}^{\mathrm{th}}$ position
• $\mathbb{C}^{n}$ : same as $\mathbb{R}^{n}$
• ${l}_{2}$ : $\{\{{\delta }_{i}(n)\}\colon i\in \mathbb{Z}\}$ , for $≔(\{{\delta }_{i}(n)\}, \{\delta (n-i)\})$ (all shifts of the Kronecker sequence)
• ${ℒ}_{2}$ : to be constructed using wavelets ...
• Say $S$ is a subspace of Hilbert space $V$ . The orthogonal complement of S in V , denoted ${S}^{\perp }$ , is the subspace defined by the set $\{x\in V\colon \perp (x, S)\}$ . When $S$ is closed, we can write $V=S\mathop{\mathrm{xor}}{S}^{\perp }$
• The orthogonal projection of y onto S , where $S$ is a closed subspace of $V$ , is $\stackrel{̂}{y}=\sum ({x}_{i}\dot y){x}_{i}$ s.t. $\{{x}_{i}\}$ is an ON basis for $S$ . Orthogonal projection yields the best approximation of $y$ in $S$ : $\stackrel{̂}{y}=\mathrm{argmin}(x\in S, (y-x))$ The approximation error $≔(e, y-\stackrel{̂}{y})$ obeys the orthogonality principle : $\perp (e, S)$ We illustrate this concept using $V=\mathbb{R}^{3}$ ( ) but stress that the same geometrical interpretation applies to any Hilbertspace.

A proof of the orthogonality principle is: $⇔(\perp (e, S), \forall i\colon e\dot {x}_{i}=0)$ $y-\stackrel{̂}{y}\dot {x}_{i}=0$

$y\dot {x}_{i}=\stackrel{̂}{y}\dot {x}_{i}=\sum ({x}_{j}\dot y){x}_{j}\dot {x}_{i}=\sum \overline{{x}_{j}\dot y}({x}_{j}\dot {x}_{i})=\sum (y\dot {x}_{j}){\delta }_{i-j}=y\dot {x}_{i}$

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers! By OpenStax By Madison Christian By OpenStax By Joanna Smithback By OpenStax By Rohini Ajay By OpenStax By John Gabrieli By OpenStax By OpenStax