# 15.4 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

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Joseph
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Lohitha
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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
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narayan
what is variations in raman spectra for nanomaterials
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Bhagvanji
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
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Rafiq
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Damian
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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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
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Rafiq
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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 ?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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