# 1.6 Hilbert spaces

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This module will provide an introduction to the concepts of Hilbert spaces.

## Hilbert spaces

A vector space $S$ with a valid inner product defined on it is called an inner product space , which is also a normed linear space . A Hilbert space is an inner product space that is complete with respect to the norm defined using the innerproduct. Hilbert spaces are named after David Hilbert , who developed this idea through his studies of integral equations. We define our valid norm using the innerproduct as:

$(x)=\sqrt{x\dot x}$
Hilbert spaces are useful in studying and generalizing theconcepts of Fourier expansion, Fourier transforms, and are very important to the study of quantum mechanics. Hilbert spacesare studied under the functional analysis branch of mathematics.

## Examples of hilbert spaces

Below we will list a few examples of Hilbert spaces . You can verify that these are valid inner products at home.

• For $\mathbb{C}^{n}$ , $x\dot y=y^Tx=\begin{pmatrix}\overline{{y}_{0}} & \overline{{y}_{1}} & \dots & \overline{{y}_{n-1}}\\ \end{pmatrix}\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ ⋮\\ {x}_{n-1}\end{array}\right)=\sum_{i=0}^{n-1} {x}_{i}\overline{{y}_{i}}$
• Space of finite energy complex functions: ${L}^{2}(\mathbb{R})$ $f\dot g=\int \,d t$ f t g t
• Space of square-summable sequences: ${\ell }^{2}(\mathbb{Z})$ $x\dot y=\sum$ x i y i

#### Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
nanocopper obvius
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, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
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