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Math Help - How does adjoint operator have to do with functionals?

  1. #1
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    How does adjoint operator have to do with functionals?

    Hi guys.
    Where can I find a proof that for every T:V\rightarrow V there exist T^* such that for all x,y\in V: <T(x),y>=<x,T^*(y)> , and that it is unique?
    I looked around in google and all I could find is a very complicated explanation about Riesz representation theorem.
    please, if someone could direct me to a simple proof that shows it, that would be great.
    but most importantly, the reason I posted this thread is that I just recently relized that it has a strong connection to functionals, and I would love to get a good understanding of it.
    so if anyone can just explain it in simple words, or post a link to a proof that shows the uniqueness of adjoint operators and how it relates to functionals, that would be great.
    I feel like I'm almost there, but I just can't get to the bottom of it.

    thanks in advanced!
    Last edited by Stormey; July 6th 2013 at 12:59 AM.
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  2. #2
    Super Member girdav's Avatar
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    Re: How does adjoint operator have to do with functionals?

    Hi,
    Can you explain a bit more the context in which your are working? For example, do you only need the case V finite dimensional?
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    Re: How does adjoint operator have to do with functionals?

    Hi, girdav,
    thanks for the help.

    first, the title should be "What does adjoint operator have to do with functionals?", of course.

    second, yes. V is a finite dimensional vector space.
    I just want to understand how is the properties of functionals used to prove the above theorem about adjoint operators (if at all).
    Last edited by Stormey; July 6th 2013 at 02:52 AM.
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  4. #4
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    Re: How does adjoint operator have to do with functionals?

    The concept of "adjoint" is defined only for functions from one vector space (more correctly one inner product space) to another:
    If A is a linear transformation from vector space U (with inner product < , >_U) to vector space V (with inner product [tex]< , >_V[tex]) then the adjoint of A, A*, is the linear transformation from V back to U such that, for any u in U, v in V, <Au,v>_V= <u, A*v>_U.

    One very important vector space is the " L^2", the space of "square integrable functions" where we can define the inner product to be \int f(x)g(x)dx. Of course, linear transformations from on function space to another are called "functionals".
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