Where to put questions about functional analysis

[Fredrik]

Functional analysis is the mathematics of linear operators between vector spaces, so it's closely related to linear algebra, which is the mathematics of linear operators between finite-dimensional vector spaces. Linear algebra is therefore a subset of functional analysis. So maybe posts should go into the linear algebra forum?

However, functional analysis is also a subset of analysis, and there is a calculus & analysis forum. So maybe that's where to post?

Then there's the fact that proofs of theorems in functional analysis rely heavily on topology. When you study functional analysis, it feels more like topology than linear algebra or analysis. So maybe posts should go into the topology & geometry forum?

People often post questions about functional analysis (about Hilbert spaces, unbounded operators, distribution theory) in the quantum physics forum too. Is that appropriate? I never report those posts to suggest a move, because I wouldn't know where to move them.

I suggest that we change something to make it clear where posts about functional analysis should be. Perhaps break off the topology forum from the geometry & topology forum and create a topology & functional analysis forum. That would be a bit weird though, since (I think) stuff about algebraic topology is probably better off in the same forum as differential geometry. (Perhaps someone who actually understands algebraic topology will have something to say about that). Another option is to turn linear & abstract algebra into linear algebra and functional analysis. I think I like that better, but then the question is where to put abstract algebra.

Now that I think about it, what about measure and integration theory? Is that set theory, or calculus/analysis? Maybe we should add a new forum called "measure theory & functional analysis" or something like that.

An alternative to changing the names of the forums is to just change their descriptions. I don't know which option I like best, but I think we should discuss the options here.

 

[Landau]

I decided to put all of my FA posts in topology and geometry (since most proofs involve methods and results from topology, usually in the most difficult part of the proof).

Yeah, functional analysis certainly uses a lot of point-set topology. But I wouldn't consider it as topology itself, I am not sure I agree with your remark about 'the most difficult part in proofs'. But the line is hard to draw, and not so interesting per se.
Just like the beginning of topology and measure theory (open sets, sigma algebras,...) are basically nothing more than set theory. But then again, in the end everything is set theory.

As you say, the calculus & analysis forum seems to be used for elementary analysis, or perhaps measure theory, mostly. Functional analysis and operator theory topics then seem out of place, and less likely to attract people interested in or knowledgeable about it.

The problem with creating a forum 'point-set topology and functional analysis' is that many analysis question can also be posted there. After all, 'convergence' and 'continuity' are inherently topological concepts. Series and the (Frechet) derivative also makes in any Banach space, making it a functional analysis topic. Etcetera...

I do agree that algebraic topology and differential geometry are more related to each other than to functional analysis. On the other hand, a good part of algebraic topology is homological algebra, which should probably in the algebra section. And what about category theory?

Basically the problem is that advanced mathematics is hugely interrelated. So I don't really have an answer I am afraid :P

 

[Studiot]

Just looking down the list of available options there are only two where I would look, were I to start a thread involving functional analysis.

Linear and Abstract Algebra

Calculus and Analysis

I think the linear algebra just has the edge in a physics forum.

The most noticeable missing heading to me is numerical analysis/methods, which could also conveniently house FA.

I don't think that FA warrants a heading of its own though.

Perhaps we could simply add "and Applications" to Linear Agebra etc?

 

[Landau]

I don't think Linear Algebra is a good choice. I think it's safe to say that the algebra part is far outweighed by the topological and analytical sides of functional analysis. Also, almost every mathematical subject can be seen as an application of linear algebra.

 

[Studiot]

Well everyone has their own opinion, that is why Fredrik started the thread. To reach a consensus.

This is a Physics Forum, so I am not coming from

"Where will/does the topic best fit into Mathematics?"

Rather "What applications do I use it for?"

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Most of the interesting real world Physics and its applications (Engineering) is non linear, although we try very hard to 'linearize' to make calculations easier.

 

[Landau]

This is a Physics Forum, so I am not coming from

"Where will/does the topic best fit into Mathematics?"

Rather "What applications do I use it for?"

While this forum has 'physics' in its title, I don't know if that means every subject should be viewed from the physics perspective. Certainly some of the members here are pure mathematicians, and a lot of interesting (in my opinion) discussions are about pure mathematics.

In my opinion the applications to physics are not really relevant for the question 'where do discussion about this (pure) mathematical subject belong?'.

Most of the interesting real world Physics and its applications (Engineering) is non linear, although we try very hard to 'linearize' to make calculations easier.

That's not really an argument for anything. The whole point of differentiating a function is 'linear approximation'. But that doesn't imply that calculus questions belong in the linear algebra forum.

 

[Fredrik]

Most questions about functional analysis don't belong in the linear algebra forum, but there are exceptions. If I had wanted to prove that the smallest subspace that contains a given subset S is the one that consists of all linear combinations of members of S, then I think the linear algebra forum would have been the right place (since there's no functional analysis forum). But a lot of stuff in functional analysis is just topology applied to vector spaces, and I think it's more appropriate to put questions about those things in the topology & geometry forum. There are exceptions to that too. For example, if the specific detail that's causing difficulties is how to prove convergence of a series, then maybe it should be in the calculus & analysis forum, or in the homework forum.

Functional analysis is definitely not an application of linear algebra. I would rather say that linear algebra is a tiny subset of functional analysis. The former is the mathematics of linear maps between finite-dimensional vector spaces, and the latter is the mathematics of linear maps between (not necessarily finite-dimensional) vector spaces. If I should pick the forum based on what the applications are, then most questions should probably be in the quantum physics forum, but I think we can all agree that would be weird.

The fact that this place is called Physics Forums is only relevant in that it brings a lot of people here who are interested in quantum mechanics. That's relevant because some of them are also interested in its mathematical foundations. (See e.g. the currently active thread Boundedness of quantum observables). The mathematical foundation of QM is functional analysis, so the fact that this place is called Physics Forums only makes it more weird that there's no obvious place to put questions about functional analysis.

 

[Landau]

Functional analysis is definitely not an application of linear algebra. I would rather say that linear algebra is a tiny subset of functional analysis. The former is the mathematics of linear maps between finite-dimensional vector spaces, and the latter is the mathematics of linear maps between (not necessarily finite-dimensional) vector spaces.

I don't quite agree. Linear algebra is not restricted to finite dimensional vector spaces. Linear algebra is just the study of vector spaces and linear maps between them. A vector space is a module over a field. A finite-dimensional vector spaces is a finitely generated module over a field. It's just that infinite dimensional vector spaces are a lot more difficult, and the algebraic notion of 'basis' - Hamel basis - is not as well-behaved. Functional analysis combines both linear algebra and analysis, to study normed vector and inner product spaces which are complete, and bounded maps between them. With tools from analysis (/topology), infinite-dimensional normed vector spaces are much more understandable.

Of course drawing these kinds of lines is not so important (and somewhat subjective). Also, functional analysis has spread its wings, and can be said to study not only normed vector spaces, but also locally convex, or even topological vector spaces, and not only continuous maps, but also for example unbounded maps between hilbert spaces.

 

[Fredrik]

The way I think about it is that functional analysis is what I said, but books and courses with "functional analysis" in the title cover the things you mentioned, because those are the parts that are best understood.

I haven't heard anyone define linear algebra to include infinite-dimensional vector spaces before.

 

[Landau]

The way I think about it is that functional analysis is what I said, but books and courses with "functional analysis" in the title cover the things you mentioned, because those are the parts that are best understood.

The way I think about it is that linear algebra is what I said, but books and courses with "linear algebra" in the title cover the things you mentioned, because those are the parts that are best understood.

I haven't heard anyone define linear algebra to include infinite-dimensional vector spaces before.

The first definition in a linear algebra course is that of a vector space. Later 'basis' is defined, then 'dimension', then 'finite dimension', and then they usually say 'we will mostly consider finite-dimensional vector spaces'. But infinite-dimensional vector spaces like $\mathbb{R}[x]$ or (more generally) $\mathbb{R}^X$, functions from a set X into R, are often discussed. Also the construction of 'the free vector space on a set', products, direct sums, tensor products of vector spaces, etc., seem to me strictly (linear) algebraic concepts, not functional analytic.

But I will stop now; I don't like discussions about (arbitrary) labels or genres, whether in music, movies, or mathematics :)