Definition talk:Functional/Real

From ProofWiki
Jump to navigation Jump to search

This seems very similar to a distribution. Is there a difference? --GFauxPas (talk) 13:34, 5 January 2017 (EST)

I believe distributions is a subset of functionals in a sense that they are linear and act on "test functions", but I am not yet educated in that area. --Julius (talk)


Well-defined-ness was only mentioned in the book by Hans Sagan, but not explained. I guess this was just a superfluous description, and could be deleted. However, if it is desirable to have something equivalent to well-defined mapping, then I at this moment I cannot help. --Julius (talk)

Domain of J

You are correct. My mistake.

Well-defined is redundant -- all mappings are "well-defined" in this context, or they are not mappings.
Removed the comment about "points", as it does not bring anything to the definition, and merged the rest of the "technical note" -- we prefer to keep that heading title for notes about how to craft the $\LaTeX$ and so on. --prime mover (talk) 14:29, 5 January 2017 (EST)

General definitions of functionals

You have asked if there are functionals mapping to spaces other than $\mathbb R$.

The answer is yes, and this is standard practice.

Definition of linear functional

  • A linear functional is a linear map from a vector space to its field of scalars.

Definition of continuous linear functional

  • A continuous linear functional is a linear map from a topological vector space to its topological field of scalars which is continuous.

More general definition

  • Given a commutative ring $R$ and an $R$-module $M$, a linear functional on $M$ is an $R$-linear map from $M$ to $R$.

Returning to the question at hand regarding (not necessarily linear) functionals, one may use the following definition:

Definition of functional

  • Given a commutative ring $R$ and an $R$-algebra $A$, a functional on $A$ is a map from $A$ to $R$.

Both the "more general definition" and the "definition of a functional" above are motivated by applications to $C^*$-algebras.

Thank you for the input. Clearly the definition has to be generalised, as I was focusing on its applications in Calculus of Variations, while the rest of contributors have been focusing on other fields. At the moment I am not planning to update the definition, at least not in the nearest future, since there is still some material left that I would like to cover, at least to some extent, but definitely it is needed. May I suggest a discussion regarding a specific page to be handled within the main discussion page of the given article? In this way any relevant information would be easily accessible without the need of auxiliary links. Julius (talk) 09:22, 19 April 2017 (EDT)
The above has been moved from there to here. --prime mover (talk) 10:24, 19 April 2017 (EDT)
Thanks for moving this to the right place, I didn't realize I was writing comments in the wrong place. While we are on this topic, another important concept is a continuous linear functional. I have added a standard definition to my comments above, in the section whose heading has been renamed to "General definitions of functionals". --M (talk) 6:36, 19 April 2017 (PST)

Refactoring in progress

So it is established that functionals can be defined that map to the complex numbers as well as to the real numbers.

There are also definitions of the concept for all sorts of other contexts as well.

We want to move on with this, so the plan is that this page be renamed to Definition:Functional/Real with a permanent redirect Definition:Real Functional.

I also propose that the subpage Definition:Functional/Weak Extremum be renamed so as not to be a subpage of Functional (should have paid attention to that when it was initially created) so it will now be Definition:Weak Extremum. If there is a need to subdefine it, then it would be Definition:Weak Extremum/Functional so as to allow different contexts of this concept, which would have a permanent redirect of Definition:Weak Extremum of Functional.

This is under way starting now. --prime mover (talk) 12:10, 25 October 2020 (UTC)

I can only agree. --Julius (talk) 15:57, 25 October 2020 (UTC)
I've gone as far as I need to on this stage of the operation. I'm going to make the effort to get my head round CoV because it's an area I haven't studied. --prime mover (talk) 16:05, 25 October 2020 (UTC)
Regarding "further research" part, for all practical purposes I had in mind functionals as multiple integrals of functions like $f : \R^n \to \R^m$. This could be generalized to $\C$ through complex modulus and real/imaginary parts. Such limitation comes from physics and geometry. I am not qualified to talk about generalized topological spaces, though. --Julius (talk) 16:20, 25 October 2020 (UTC)