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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

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
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Jyoti Reply
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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
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Brian Reply
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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
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LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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Santosh
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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
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Stoney Reply
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Kyle
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Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
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Ernesto
Introduction about quantum dots in nanotechnology
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Loga
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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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