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How to define an 'infinite dimensional integral'

  1. Jan 9, 2007 #1
    Hello, my question is how could we define integration on infinite dimensional spaces?, my idea is, let be the multiple integral.

    [tex] \int_{V}dVf(X) [/tex] where [tex] X=(x_1 ,x_2 , x_3 ,.....,x_n ) [/tex]

    then i define a family of trial functions, in my case they are just 'step functions' so [tex] H(X)=H(x_1) H(x_2 ) H(x_3 ).....H(x_n) [/tex] and define a some kind of axiomatic integral for htem (i don't know how unfortunately) then i try to apply integration by parts so integral hold to and make n -> infinite so we define an infinite dimensional integral.

    By the way, is there an analogue to Euler-Mc Laurin sum formula for infinite dimensional spaces?
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  3. Jan 9, 2007 #2


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  4. Jan 9, 2007 #3


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    Unfortunately the second article is not free, unless you have a paid subscription to Elsevier.

  5. Jan 9, 2007 #4
    But the infinite dimensional derivative (functional derivative) can be defined for a functional in 2 ways:

    [tex] \frac{F[\phi +\epsilon \delta (x-y)]-F[\phi]}{\epsilon} [/tex] or

    [tex] \frac{dF[\phi +\epsilon (x-y)]}{d\epsilon} [/tex]

    for epsilon tending to 0, and it yields to Euler-Lagrange equation, the question is why can't we define the integral by means perhaps of the sum, with epsilon tending to 0 in the form?

    [tex] \sum_{n=0}^{\infty}\epsilon F[\phi +n\epsilon \delta (x-y)] [/tex] or if we denote the functional derivative operator [tex] \delta [/tex] then its inverse is just the functional integral operator .
    Last edited: Jan 9, 2007
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