How to define an 'infinite dimensional integral'

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In summary, the conversation discusses defining integration on infinite dimensional spaces. The idea is to use multiple integrals and a family of trial functions, such as step functions, to define an infinite dimensional integral. However, the problem arises with defining the topology in infinite dimensions. The conversation also mentions the possibility of using an analogue to the Euler-McLaurin sum formula for infinite dimensional spaces, and the definition of the infinite dimensional derivative (functional derivative) in two ways.
  • #1
Kevin_spencer2
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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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  • #2
One problem you will have is that the various plausible "metrics", which give exactly the same topology in finite dimensions and so are equivalent, give different topologies in infinite dimensional space. That is, before you can define integrals or derivatives in infinite dimensions, you will need to specify the topology.

You might want to look at this
http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=528446
or this
http://adsabs.harvard.edu/abs/1979PhLA...73..287B
 
  • #3
Unfortunately the second article is not free, unless you have a paid subscription to Elsevier.

Daniel.
 
  • #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 .
 
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What is an infinite dimensional integral?

An infinite dimensional integral is a mathematical concept that involves integrating a function over an infinite number of variables or an infinite range of values. Unlike traditional integrals, which involve a finite number of variables and values, infinite dimensional integrals can be used to solve complex problems in areas such as physics, mathematics, and engineering.

How is an infinite dimensional integral defined?

The precise definition of an infinite dimensional integral may vary depending on the context and the specific problem being solved. In general, it involves taking a limit of a sequence of integrals with an increasing number of variables or values. This limit is often represented as a sum or a product, and the resulting value is known as the integral.

What are some applications of infinite dimensional integrals?

Infinite dimensional integrals have many applications in various fields, including quantum mechanics, statistical mechanics, and functional analysis. They are used to calculate probabilities, solve differential equations, and study complex systems with an infinite number of degrees of freedom.

How do you evaluate an infinite dimensional integral?

Evaluating an infinite dimensional integral can be a challenging task and may require advanced mathematical techniques such as Fourier analysis, functional analysis, or complex analysis. In some cases, the integral may have an analytical solution, but in most cases, it can only be approximated numerically using algorithms and computer programs.

What are some common challenges when dealing with infinite dimensional integrals?

One of the main challenges when working with infinite dimensional integrals is the lack of a well-defined measure or a suitable coordinate system. This can make it difficult to interpret the integral and determine its convergence. Another challenge is the complexity of the integrand, which may involve complex functions or distributions that are not well-behaved. Additionally, numerical approximations can introduce errors and may require a significant amount of computing power and resources.

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