# Extremely infinite dimensional calculus

1. Aug 29, 2008

### jostpuur

For functions

$$f:\mathbb{R}^n\to\mathbb{R}$$

we have derivatives

$$\frac{\partial}{\partial x_k} f,$$

and for functionals

$$F:\mathcal{H}\to\mathbb{R},\quad\quad\quad \mathcal{H}\subset \mathbb{R}^{\mathbb{R}^n}$$

we have functional derivatives

$$\frac{\delta}{\delta f(x)} F.$$

But if we have a linear form defined on a space of functionals,

$$\mathcal{F}:\mathcal{Z}\to\mathbb{R},\quad\quad\quad \mathcal{Z}\subset \mathbb{R}^{\mathcal{H}},$$

then what's the name for this

$$\frac{\mathcal{D}}{\mathcal{D} F(f)} \mathcal{F}?$$

Did I manage giving it a logical notation at least? In any case, I have no idea what it should be called.

2. Aug 29, 2008

These ideas may not be what you are after, but google both the Gateux and the Frechet derivatives. Both of these are generalizations of multivariable derivatives to operators on function spaces.

3. Aug 29, 2008

### jostpuur

Actually I was explaining unnecessarily much about what I was doing. It would suffice to know what linear forms on the spaces of functionals are called.

4. Aug 29, 2008

### DarMM

In any context I have seen they are simply called "Functionals of Functionals" or "function of functionals". An example of such an object is the action in quantum field theory. The path-integral in qft is in fact an integral of such an object over a space of functionals.

5. Aug 29, 2008

### jostpuur

Unfortunately a "functional of functional derivative" would sound dumb.

6. Aug 29, 2008

### morphism

Why would there be a need to call functionals of functionals anything other than simply 'functionals'? After all, in a space of functionals, the functionals themselves are the 'vectors.'

7. Aug 29, 2008

### jostpuur

I was just going to explain that the linear functionals are always linear forms, but the term "functional" is not redundant, because it is used when the domain is a space of functions, where as the term "form" would be preferred in more general context when there is no need to emphasize what the domain is. But then I checked the Wikipedia page http://en.wikipedia.org/wiki/Linear_functional and it seems that I could have been having a wrong belief with this terminology convention. Wikipedia treats these terms synonymous. Could it be, that I was thinking like this because the term "functional" is in reality anyway still mostly used only for the space of functions?

In principle we could just manage with the word "mapping", but we have lot of words like "function", "form", "operator", "functional", because it brings clarity to call certain kind of mappings with certain names. It would not be impossible to consider these all synonymous, and then cause lot of confusion, but I would prefer sticking with conventions whenever they exist.

8. Aug 29, 2008

### morphism

In my experience the phrase "linear functional" is much more common than "linear form." Actually, I don't think I've ever seen the phrase "linear form" in a modern, mainstream analysis text.

9. Aug 29, 2008

### jostpuur

If I took a n component vector v and said that it is the functional derivative of the linear map

$$\mathbb{R}^n\to\mathbb{R},\quad x\mapsto v\cdot x$$

surely somebody would protest against this terminology?

10. Aug 29, 2008

### DarMM

I just wanted to say that the calculus of such objects is very difficult to make rigorous. In fact renormalization in quantum field theory basically relates to these difficulties.

11. Aug 29, 2008

### Ben Niehoff

Why not call them "superfunctionals"?

12. Aug 29, 2008

### jostpuur

If nobody else is calling them with that name, that would be one reason to avoid this name. Not a kind of reason that would totally prevent the use of this name, but a reason anyway. Who would have self-confidence to introduce new terminology into mathematics??

13. Aug 30, 2008

### Ben Niehoff

Bah, I make up new terminology, and new notation, all the time. What's important is that you explain it at the beginning so people know what you're referring to. There are all sorts of terminology and notation conflicts in mathematics already, so you always have to explain yourself anyway.

14. Aug 30, 2008

### mhill

The Question is , if we can define a Frechet or Gateaux derivative, why could not define a Gateaux integral to introduce a generalization of a Functional integral ??

15. Sep 5, 2008

### jostpuur

It was a mistake to start speaking about linear mappings in this context, because actually linearity is not really essential. For example,

$$x\mapsto \int dt\;\big(\frac{1}{2}(\dot{x}(t))^2 - V(x(t))\big)$$

is not a linear mapping, but it is called a functional in the context of the Hamilton's principle. A functional derivative, in an infinite dimensional context, is not necessarily taken of a linear functional, but can be taken of a non-linear functional too.

http://en.wikipedia.org/wiki/Functional_(mathematics)

16. Sep 5, 2008

### Ben Niehoff

The difficulty is in defining a measure on infinite-dimensional space. For a finite-dimensional space, you can simply use N iterated integrals. But for infinite dimensions, you have to take the limit as N goes to infinity, and there might not be a consistent way to define this limit and show that it exists.