Duality Pairing and Functionals

[maros522]

Hello all,

does anybody know what means duality pairing in connection with functional. For example lim_{E$$\rightarrow$$0}$$\frac{\partial}{\partialE}$$F(u+Ev)=<DF(u),v>. Where F is functional F:K$$\rightarrow$$R.

Thank You for answers.

 

[nirax]

please write with proper formatting .. it is not possible to guess what you mean ...

 

[maros522]

Hello, I find definition of duality pairing in book
http://books.google.cz/books?id=zTV...onepage&q=duality pairing functional&f=false"
The part of interest is as jpg in attachments - dualitypairing1.jpg
But in book Contact problem in elasticity from Oden and Kikuchi is definition like in dualitypairing2.jpg.
In dualitypairing2.jpg is used as functional gradient of functional F at u. I don't understand how it is meaned. If g is part of V' we write g(v)=<g,v>: in this the g is functional. But in dualitypairing2.jpg is DF(u), which is gradient of F at u. This DF(u) is still functional or is it a value.

 

[nirax]

nobody will bother replying to you if you don't make any effort to clarify what u r asking.

 

[zhentil]

DF(u) is a functional. Think of it this way: the gradient of a function takes a point and gives you back a vector. The inner product on euclidean space allows you to transform that vector into a function.

 

[maros522]

Thank you for posting messages.
Nirax: the question was "This DF(u) is still functional or is it a value?" I forget to add ?.
Zhentil: The inner product on euclidean space is dot product of two vectors. So the result will be real number. How do you think it?