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Product of linear functional

  1. Jul 15, 2007 #1
    Let be a set of LInear functionals [tex] U_{n}[f] [/tex] n=1,2,3,4,.........

    so for every n [tex] U_{n}[\lambda f+ \mu g]=\lambda U_{n}[f]+\mu U[g] [/tex] (linearity)

    the question is if we can define the product of 2 linear functionals so

    [tex] U_{i}U_{j}[f] [/tex] makes sense.
     
  2. jcsd
  3. Jul 15, 2007 #2

    radou

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    You can define the product of linear functionals as composition, so you have an algebra of functionals.
     
  4. Jul 15, 2007 #3

    HallsofIvy

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    Defining an "ordinary" product, that is as the product of the results of the functionals would destroy linearity.

    Unfortunately, since a "functional" maps functions to numbers, the composition of two functionals does not exist.
     
  5. Jul 15, 2007 #4

    radou

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    Of course, I didn't think about that. :uhh:
     
  6. Jul 15, 2007 #5

    lurflurf

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    One could procede as in multilinear algebra/tensor calculus and define an outer product.

    Thus let u,v be functionals
    u,v:V->F
    (uv)f=(vf)u

    One might say the product between two linear functionals is a bilinear functional.
     
  7. Jul 16, 2007 #6

    matt grime

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    You can trivially define the product (as in multiplication) of two linear functionals. It just isn't a linear functional. A linear function is in particular a C/R/F valued function, so it lies in the algebra of functions, as radou sort of said.
     
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