Extremely infinite dimensional calculus

[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.

 

[statdad]

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.

 

[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.

 

[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.

 

[jostpuur]

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

 

[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.'

 

[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.

 

[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.

 

[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?

 

[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.

 

[Ben Niehoff]

Why not call them "superfunctionals"?

 

[jostpuur]

Why not call them "superfunctionals"?

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??

 

[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.

 

[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 ??

 

[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)

In mathematics, a functional is traditionally a map from a vector space to the field underlying the vector space, which is usually the real numbers. In other words, it is a function that takes a function as its argument or input and returns a scalar.

A special kind of such functionals, linear functionals, gives rise to the study of dual spaces.

 

[Ben Niehoff]

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 ??

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.