functions of uncountably many variables

[imfatandugly]

is there such a thing? If there is could someone point me in the right direction

[DarkMagus]

A functional is essentially a function of a function, which is sort of what you're talking about.

http://en.wikipedia.org/wiki/Functio...mathematics%29

I suppose you could also consider a function of an infinite sequence of discrete variables, but I don't know the name of such a thing.

[rufus]

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is there such a thing? If there is could someone point me in the right direction

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If you're familiar with them, the derivative and integral are examples.

[T50_Omaha8]

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is there such a thing?

[/ QUOTE ] GDP growth, interest rates, industrial output, stock prices, etc etc etc

Do you mean uncountably infinite in a strict mathematical sense, as in having no equivalence relation to the natural numbers?

If so, I don't understand how the derivative or integral satisfy this property.

[imfatandugly]

Yes uncountably many independent variables as in infinately many but not in one to one correspondence qith the natural numbers. Yes i did not see how the der or integral were examples either although I can see that the der/int take into account infinately many values ( but they are not variables or independent). Right?

[rufus]

It's possible to think of real functions as (uncountably) infinite-dimensional vectors. As such any function that maps real functions to something else is a function of infinitely many variables.

Another (admittedly trivial) example of a function of infinitely many variables is a function that takes any ordered set of real numbers with the cardinality of the continuum, and is constant.

[imfatandugly]

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It's possible to think of real functions as (uncountably) infinite-dimensional vectors. As such any function that maps real functions to something else is a function of infinitely many variables.


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so the function g(x)=x+1 is actually a function of infinately many variables, not one? lame.

[rufus]

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so the function g(x)=x+1 is actually a function of infinately many variables, not one? lame.

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Not really. But you could say that g(f(x))=f(x)+1 is a function of infinitely many variables.

Functions that operate on real functions rather than operating on real numbers are effectively functions of uncountably many variables.

I don't know that you'd ever want to do it but the characteristic function of continuity g:f->{0,1} which is 1 if f is continuous everywhere, and 0 otherwise.

[jason1990]

Consider the vector v = (3,1,7,9). Normally, we use subscript notation to denote its components:

v_1 = 3
v_2 = 1
v_3 = 7
v_4 = 9

But sometimes we use parentheses instead:

v(1) = 3
v(2) = 1
v(3) = 7
v(4) = 9

When we use parentheses, it makes it clear that the vector v is really just a function whose domain is {1,2,3,4}. Conversely, any real-valued function whose domain is {1,2,3,4} is really just a vector in R^4.

Now consider the infinite sequence a = {2,4,8,16,32,...}. This can be regarded as a vector with a countably infinite number of components. Normally, we use subscript notation for the elements of the sequence:

a_1 = 2
a_2 = 4
...
a_n = 2^n
...

But sometimes we use parentheses instead:

a(1) = 2
a(2) = 4
...
a(n) = 2^n
...

When we use parentheses, it makes it clear that the sequence a is really just a function whose domain is {1,2,3,...}. Conversely, any function whose domain is {1,2,3,...} is really just an infinite sequence. In other words, an infinite sequence can be regarded either as a vector with countably many components (one component for every natural number) or as a function whose domain is {1,2,3,...}. Mathematically, they are the same thing. There is no mathematical difference between functions and vectors. A function is just a vector that has one component for every point in its domain.

Extending this reasoning, any function whose domain is R is really just a vector that has one component for each real number. That is, it is a vector that has uncountably many components.

A function of 4 variables is a function whose domain consists of vectors in R^4. That is, a function of 4 variables is a function of functions on {1,2,3,4}. A function of countably many variables is a function whose domain consists of sequences. That is, a function of countably many variables is a function of functions on {1,2,3,...}. Likewise, an example of a function of uncountably many variables would be a function of functions on R. rufus has given several examples.

[imfatandugly]

Thanks. Good explanation.

What would happen to Q if you adjoined uncountably many variables to it? Anything interesting?