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Duality pairing

  1. Aug 26, 2009 #1
    Hello all,

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

    Thank You for answers.
  2. jcsd
  3. Aug 27, 2009 #2
    please write with proper formatting .. it is not possible to guess what you mean ...
  4. Aug 27, 2009 #3
    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 dont 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.

    Attached Files:

    Last edited by a moderator: Apr 24, 2017
  5. Aug 28, 2009 #4
    nobody will bother replying to you if you dont make any effort to clarify what u r asking.
  6. Sep 2, 2009 #5
    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.
  7. Sep 4, 2009 #6
    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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