# 1.13 Infinite-dimensional hilbert spaces

 Page 1 / 1
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 $\left(X,R,+,·\right)$ be a vector space. An infinite sequence of orthonormal vectors $\left\{{e}_{1},{e}_{2},...\right\}\subseteq X$ is said to be a complete orthonormal sequence (CONS) in $X$ if for every $x\in X$ there exists a sequence ${\alpha }_{1},{\alpha }_{2},...\in R$ such that $x={\sum }_{i}{\alpha }_{i}{e}_{i}$ .

For the sake of concreteness, an infinite sum is defined as $x={\sum }_{i=1}^{\infty }{\alpha }_{i}{e}_{i}={lim}_{n\to \infty }{\sum }_{i=1}^{n}{\alpha }_{i}{e}_{i}$ . It is easy to see that ${\alpha }_{i}=〈x,,,{e}_{i}〉$ .

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, $R=\mathbb{C}$ , a CONS is given by ${e}_{k}\left(t\right)=\frac{1}{\sqrt{2\pi }}{e}^{jkt}$ for $k=0,±1,±2,...$ . These vectors are orthonormal:

$〈{e}_{k},,,{e}_{l}〉={\int }_{0}^{2\pi }\frac{1}{2\pi }{e}^{jkt}{e}^{-jlt}dt={\int }_{0}^{2\pi }\frac{1}{2\pi }{e}^{j\left(k-l\right)t}dt=\left\{\begin{array}{cc}1\hfill & k=l\hfill \\ 0\hfill & k\ne l\hfill \end{array}\right)$

The coefficients are given by

${c}_{k}=〈x,,,{e}_{k}〉={\int }_{0}^{2\pi }x\left(t\right){e}^{-jkt}dt,$

and we obtain $x={\sum }_{k}{c}_{k}{e}_{k}$ . This is the sequence behind the Fourier series representation.

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

${e}_{k}\left(t\right)=\frac{1}{\sqrt{2B}}\mathrm{sinc}\left(2,B,\left(t,-,\frac{k}{2B}\right)\right),$

where $\mathrm{sinc}\left(t\right)=\left(sin\left(\pi t\right)\right)/\left(\pi t\right)$ . It is possible to show that the functions are orthogonal to each other, i.e.,

$〈{e}_{k},,,{e}_{l}〉={\delta }_{k,l}\left\{\begin{array}{cc}1\hfill & k=l\hfill \\ 0\hfill & k\ne l\hfill \end{array}\right).$

If $x$ is bandlimited, then it follows that $x\left(t\right)={\sum }_{k}{c}_{k}{e}_{k}\left(t\right)$ , with ${c}_{k}=〈x,,,{e}_{k}〉=x\left(k/\left(2B\right)\right)$ . 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 $x\in X$ :

${\parallel x\parallel }^{2}=\sum _{i}|〈x,,,{e}_{i}〉{|}^{2}=\sum _{i}{|{c}_{i}|}^{2}.$

The sequence $\left\{{e}_{i}\right\}$ is CONS if and only if

$x=\sum _{i=1}^{\infty }〈x,,,{e}_{i}〉{e}_{i}=\underset{n\to \infty }{lim}\sum _{i=1}^{n}〈x,,,{e}_{i}〉{e}_{i}=\underset{n\to \infty }{lim}{x}_{n},$

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

${\parallel x\parallel }^{2}=\underset{n\to \infty }{lim}\parallel x-{x}_{n}{\parallel }^{2}+\underset{n\to \infty }{lim}\parallel {x}_{n}{\parallel }^{2}=0+\underset{n\to \infty }{lim}\sum _{i=1}^{n}|〈x,,,{e}_{i}〉{|}^{2}=\sum _{i=1}^{\infty }{|〈x,,,{e}_{i}〉|}^{2}.$

Theorem 2 Let $X$ be a Hilbert space with a CONS $\left\{{e}_{1},{e}_{2},...\right\}$ . Then for any $x,y\in X$ , Parseval's relation holds: $〈x,,,y〉={\sum }_{i}〈x,,,{e}_{i}〉\overline{〈y,,,{e}_{i}〉}$ .

Using the CONS, we can write the partial sums ${x}_{n}={\sum }_{i=1}^{n}〈x,,,{e}_{i}〉{e}_{i}$ and ${y}_{n}={\sum }_{i=1}^{n}〈y,,,{e}_{i}〉{e}_{i}$ . We then have

$\begin{array}{cc}\hfill |〈{x}_{n},,,{y}_{n}〉-〈x,,,y〉|& =|〈{x}_{n},,,y〉-〈{x}_{n},,,y〉+〈{x}_{n},,,{y}_{n}〉-〈x,,,y〉|\hfill \\ & =|〈{x}_{n},,,{y}_{n},-,y〉+〈{x}_{n},-,x,,,y〉|\hfill \\ & \le |〈{x}_{n},,,{y}_{n},-,y〉|+|〈{x}_{n},-,x,,,y〉|\hfill \\ & \le \parallel {x}_{n}\parallel \parallel {y}_{n}-y\parallel +\parallel {x}_{n}-x\parallel \parallel {y}_{n}\parallel \hfill \end{array}$

Letting $n\to \infty$ we have that the upper bound goes to zero, and therefore as $n\to \infty$ , we have $〈{x}_{n},,,{y}_{n}〉\to 〈x,,,y〉$ . Therefore,

$\begin{array}{cc}\hfill 〈x,,,y〉& =\underset{n\to \infty }{lim}〈{x}_{n},,,{y}_{n}〉=\underset{n\to \infty }{lim}\sum _{i=1}^{n}\sum _{j=1}^{n}〈〈x,,,{e}_{i}〉,{e}_{i},,,〈y,,,{e}_{j}〉,{e}_{j}〉\hfill \\ & =\underset{n\to \infty }{lim}\sum _{i=1}^{n}\sum _{j=1}^{n}〈x,,,{e}_{i}〉\overline{〈y,,,{e}_{j}〉}〈{e}_{i},,,{e}_{j}〉=\underset{n\to \infty }{lim}\sum _{i=1}^{n}〈x,,,{e}_{i}〉\overline{〈y,,,{e}_{i}〉}\hfill \\ & =\sum _{i=1}^{\infty }〈x,,,{e}_{i}〉\overline{〈y,,,{e}_{i}〉}.\hfill \end{array}$

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
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
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!      By Subramanian Divya    