



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:
$(x)=\sqrt{x\cdot x}$
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
$\mathbb{C}^{n}$ ,
$$x\cdot y=y^Tx=\begin{pmatrix}\overline{{y}_{0}} & \overline{{y}_{1}} & \dots & \overline{{y}_{n1}}\\ \end{pmatrix}\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ \vdots \\ {x}_{n1}\end{array}\right)=\sum_{i=0}^{n1} {x}_{i}\overline{{y}_{i}}$$
 Space of finite energy complex functions:
${L}^{2}(\mathbb{R})$
$$f\cdot g=\int \,d t$$∞
∞
f
t
g
t
 Space of squaresummable sequences:
${\ell}^{2}(\mathbb{Z})$
$$x\cdot y=\sum $$∞
∞
x
i
y
i
Questions & Answers
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
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 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?
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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?
how did you get the value of 2000N.What calculations are needed to arrive at it
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