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Describes conditions for local optimization in Hilbert Spaces

We also must define the notion of an extremum in an arbitrary normed space.

Definition 1 Let f be a real-valued functional defined on Ω X where X is a normed space. A point x 0 Ω is a local/relative minimum of f on Ω if f ( x 0 ) f ( x ) for all x Ω such that x - x 0 < ϵ for some ϵ > 0 .

Definition 2 Let f be a real-valued functional defined on Ω X where X is a normed space. A point x 0 Ω is a local maximum of f on Ω if f ( x 0 ) f ( x ) for all x Ω such that x - x 0 < ϵ for some ϵ > 0 .

Definition 3 Let f be a real-valued functional defined on Ω X where X is a normed space. A point x 0 Ω is a local strict minimum of f on Ω if f ( x 0 ) < f ( x ) for all x Ω such that x - x 0 < ϵ for some ϵ > 0 .

Definition 4 Let f be a real-valued functional defined on Ω X where X is a normed space. A point x 0 Ω is a local strict maximum of f on Ω if f ( x 0 ) > f ( x ) for all x Ω such that x - x 0 < ϵ for some ϵ > 0 .

It turns out the notion of a gradient is intrinsically linked to the directional derivatives we have introduced.

Definition 5 Let X be a Hilbert space and f : X R . If f is a Fréchet differentiable functional, then for each x X there exists a vector in X such that δ f ( x ; h ) = h , f ( x ) for all h X ; the vector f ( x ) is called the gradient of f at x , and can be written as a functional f : X X .

This definition can be seen to correspond to an application of the Riesz representation theorem to the Fréchet derivative δ f ( x ; h ) , which is a linear bounded functional on h .

Example 1 We know now that:

δ f ( x ; h ) = h , f ( x ) .

By the Cauchy-Schwarz Inequality, we have:

| δ f ( x ; h ) | = | h , f ( x ) | h f ( x ) .

If h = f ( x ) then δ f ( x ; h ) is maximized.

Example 2 Recall that if f : R n R , then

f ( x ) = f x 1 f x 2 f x n .

Theorem 1 Let f : X R have a Gâteaux differential on X . A necessary condition for f to have an extremum at x 0 X is that δ f ( x 0 ; h ) = 0 for all h X . Alternatively, if X is a Hilbert space, we can write h , f ( x 0 ) = 0 for all h X , which implies f ( x 0 ) = 0 .

Suppose x 0 is a local minimum. Then there exists ϵ > 0 such that if x - x 0 < ϵ then f ( x 0 ) f ( x ) . Fix h 0 and let θ = ϵ h . Next, consider x = x 0 + α h . For α ( - θ , θ ) :

  • If α > 0 then f ( x 0 + α h ) - f ( x 0 ) α 0 , and therefore δ f ( x 0 ; h ) 0 .
  • If α < 0 then f ( x 0 + α h ) - f ( x 0 ) α 0 , and therefore δ f ( x 0 ; h ) 0 .

Therefore, δ f ( x 0 ; h ) = 0 for arbitrary nonzero h . Now since δ f ( x ; h ) is linear on h we must have δ f ( x ; h ) = 0 for h = 0 . Therefore, the equality is true for all h X .

Definition 6 A point at which δ f ( x ; h ) = 0 for all h X is called a stationary point of f .

Questions & Answers

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
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
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Signal theory. OpenStax CNX. Oct 18, 2013 Download for free at http://legacy.cnx.org/content/col11542/1.3
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