# Integration/Differentiation on abstract spaces

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## Main Question or Discussion Point

Hi, everyone:

I was wondering if it makes sense to define a theory of integration
in abstract spaces (i.e., spaces other than IR^n, or homeomorphs),
and, if so, how to do it (and wether we can then define a theory of
differentiation). If so, do we need to define a measure, and what
measure do we use? (with the theory of integration satisfying
"reasonable" properties, like monotonicity, additivity, etc.,
similar to those of the Riemann integral, if not the Lebesgue integral.).
It would seem like the Borel Algebra of X would do the job for a measure, but I
am not sure.

This (together with a triple espresso) is what got me started :

I have been reading J.Rotman's homological algebra book.

In it, he defines homology as an equiv. class of curves C: X->IR^2,

X a top. space, and we consider functions f on X satisfying dP/dx=dQ/dy

(this seems strange to start with: can we define derivatives on X if X is

not at least a normed space? *) .

Specifically, two curves C,C' are equivalent, if :

Integral_ (C)f =Integral_(C')f ,i.e. , if Integral_(C-C')f=0 ,

where we define C-C' as a formal sum of curves:

Integral_(C-C')f =Integral_Cf -Integral_C'(f)

so -C' is the curve C' , with reversed orientation.

* Related to this issue, I was confused over a proof I saw of

the inverse function theorem, when we were working on a f.dim vector

space V/IR --not assumed to be a normed space. ( And, from what I

could tell, we were not giving V charts as a manifold thru :

(v1,...,vn)-->(f1,...,fn) , using the isomorphism between V and IR^n .

Can we define derivatives in spaces without having a norm?

Thanks.

## Answers and Replies

Related Differential Geometry News on Phys.org
Reb
The http://en.wikipedia.org/wiki/G%C3%A2teaux_derivative" [Broken] derivative.

Integration on Banach spaces is a more delicate subject. There is however an approach parallel to the construction of the Lebesque integral.
A taste of it http://www.worldscibooks.com/etextbook/5905/5905_chap1.pdf" [Broken]

(ps. no copyright infridgement)

Last edited by a moderator:
You can do both. Integration requires a Riemannian metric, which allows you to define a volume form. Differentiation requires a covariant derivative, if you plan on going past the first derivative.