Duality Pairing and Functionals

AI Thread Summary
Duality pairing in the context of functionals relates to the interaction between a functional and its gradient. The discussion highlights a specific formula involving the limit of a functional's derivative, suggesting that the gradient DF(u) is indeed a functional. There is clarification that while DF(u) can be viewed as a functional, its application in an inner product context yields a real number. The conversation emphasizes the importance of clear communication when discussing complex mathematical concepts. Understanding these relationships is crucial for grasping the underlying principles of duality pairing in functional analysis.
maros522
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Hello all,

does anybody know what means duality pairing in connection with functional. For example limE\rightarrow0\frac{\partial}{\partialE}F(u+Ev)=<DF(u),v>. Where F is functional F:K\rightarrowR.

Thank You for answers.
 
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please write with proper formatting .. it is not possible to guess what you mean ...
 
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.
 

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