Talk:Functional analysis

Disambiguation Page?

Functional Analysis is also used to refer to functionalism in Sociology and Sociocultural Anthropology. Judging from that other person's comment about Applied Behavior Analysis in Psychology, it would be easier to navigate and more informative if there was a disambiguation page (or at least one of those things at the top that says "redirected from disambiguation page") for people looking for information in those fields. I'm new to wiki and aren't quite sure how to do that though... help? 192.246.226.152 (talk) 21:27, 11 March 2007 (UTC)

Functional analysis (sometimes "functional harmonic analysis" or "harmonic analysis") in music theory refers to the formal analysis of functional harmony, which doesn't yet have a dedicated article but AFAIK is a widely used term within the field. Confusingly, harmonic analysis is another unrelated concept in mathematics that's sometimes used to mean the same thing. Ormaaj (talk) 23:21, 21 December 2009 (UTC)

Hahn-Banach theorem

The error was saying that the Hahn-Banach theorem requires the axiom of choice when it doesn't. Cwzwarich (talk) 02:25, 4 December 2005 (UTC)

Are you sure? At Hahn-Banach theorem they say that it requires Zorn's lemma, which is the same as the axiom of choice. Oleg Alexandrov (talk) 02:29, 4 December 2005 (UTC)

Yes, I am sure. Two references are the two papers of Pincus from 1972: "Independence of the prime ideal theorem from the Hahn Banach theorem" - Bull. Amer. Math. Soc. 78, 766-770, and "The strength of the Hahn-Banach theorem", from the Proc. of the Victoria Symp. on Nonstandard Analysis, Lecture Notes in Math. 369, Springer, 203-248. Thanks for letting me know about the other page. I will correct it. Cwzwarich (talk) 02:36, 4 December 2005 (UTC)

According to my lecturer, for separable real normed vectors spaces you don't need the Axiom of Choice, but for complex non separable spaces you do. 80.6.80.51 (talk) 22:58, 8 February 2007 (UTC)

It really depends what you mean by "need". Hahn-Banach is strictly weaker than the axiom of choice, so it's possible to use other, weaker axioms to prove Hahn-Banach, however, it's not possible to prove Hahn-Banach with ZF axioms alone - i.e. without the axiom of choice or something a lot like it. By way of analogy, you need the axiom of choice to prove Hahn-Banach in the same way you need a hammer to get into a nut; you could also do it with a nutcracker, or maybe just hit it with a rock, but you sure as hell can't open it with your bare hands. 128.240.229.3 (talk) 14:04, 26 June 2007 (UTC)

It says "... the Hahn-Banach theorem, usually proved using axiom of choice, although the strictly weaker Boolean prime ideal theorem suffices." It does not make sense since if the Boolean prime ideal theorem suffices, it suffices. It does not matter what people do usually on blackboard or in textbooks, the underlying mathematics remains the same. Temur (talk) 10:47, 1 March 2008 (UTC)

The boolean prime ideal theorem is taken as a replacement of the axiom of choice in some axiomatic systems for set theory. Hence, literally, the underlying mathematics will be quite different. Of course, if we are using ZFC, then there is no reason not to use the axiom of choice (in which case, there is no difference whether you use choice or the BPI). Silly rabbit (talk) 14:24, 1 March 2008 (UTC)

Ok. Thanks for the clarification. Temur (talk) 20:51, 1 March 2008 (UTC)

Axiom of choice is strictly weaker than the Boolean prime ideal theorem ?

In the "Foundations of mathematics considerations" section, there is a statement:

"Many very important theorems require the Hahn-Banach theorem, which relies on the axiom of choice that is strictly weaker than the Boolean prime ideal theorem."

However, I got confused when I followed the link to the "Boolean prime ideal theorem". On that page, it seems like it is the other way round, fx:

"Instead, some of the statements turn out to be equivalent to the axiom of choice (AC), while others, like the Boolean prime ideal theorem, represent a property that is strictly weaker than AC".

I don't know which is true, but to me it looks like there is some kind of contradiction. Alkelele (talk) 20:23, 15 May 2007 (UTC)

BPI is strictly weaker than AxC, as its article says. Hahn-Banach is equivalent to BPI, iirc. I'll change it, probably that's what the author meant, but wrote it wrong. John Z (talk) 03:15, 16 May 2007 (UTC)

Soft analysis

Soft analysis redirects here, but nothing explains why or how this is considered "soft". The phrase itself is never used in the article.--Prosfilaes (talk) 17:09, 6 January 2010 (UTC)

It's an imprecise slang term, and the redirection reflects only one of its uses. [1] [2] [3] Functional analysis includes both some soft and some hard analysis. [4] 92.231.231.121 (talk) 01:40, 9 January 2011 (UTC)

Anna Johnson Pell Wheeler

I have removed the link added with the summary "See also: link to Anna Johnson Pell Wheeler (famous for theories on func. analysis". From Mac Tutor: "The direction of Anna Wheeler's work was much influenced by Hilbert. Under his guidance she worked on integral equations studying infinite dimensional linear spaces. This work was done in the days when functional analysis was in its infancy and much of her work has lessened in importance as it became part of the more general theory." That does not exactly translate as famous. If any links to persons were appropriate here, other names come to mind, such as Stefan Banach, etc. Jmath666 (talk) 00:09, 9 May 2010 (UTC)

Introduction: less is more?

The new introduction (since December 21st) is rather long (almost 400 words), and includes a number of run-on sentences. I'd be in favour of reverting to an earlier version, unless someone has a better idea. Jowa fan (talk) 03:07, 22 December 2010 (UTC)

Set of functions over sets

it's great, but I thought it could read a little different, when you say the structured set, it's not entirely clear if you mean the set of functions or the only sets you previously mentioned, the underlying sets over which the functions are defined. Brady, apr 3 2011 —Preceding unsigned comment added by 166.205.143.35 (talk) 00:32, 4 April 2011 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Functional analysis/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

This article doesn't do enough to emphasise the incredible importance of functional analysis. It would also benefit from a historical overview of the subject and its connection to quantum mechanics. It does a nice job of presenting major results in a clear fashion. shotwell 08:38, 8 October 2006 (UTC)

Last edited at 12:18, 15 April 2007 (UTC). Substituted at 02:07, 5 May 2016 (UTC)

Huge lead restructure

I have an intermediate level of mathematical sophistication, having once completed half a math degree, and having worked in STEM thereafter.

The pre-existing lead was simply not helping me form a quick map of how functional analysis and the wavelet encoding interrelated. The problem was common to many math pages I've seen before, where the first editor arrived with an extremely formal mandate, and the page ended up becoming brittle and (apparently) correct, but not inviting to further work.

I've almost certainly taken this too far now in the direction of accessibility and done damage to the precise formal claims. I tried, but I surely failed here.

So as not to leave this article in different persistent state of neglect, I aggressively flagged my guesswork with expert-needed and speculation.

My agenda here was simply to learn the material, and I made these edits passing through. Consider the torch passed. — MaxEnt 19:31, 21 August 2020 (UTC)

I seriously don't understand what happened here, even the definition of functional analysis is wrong and at best unfamiliar. I.e. the original stuff was better [[5]]. Trying to learn Wavelets with respect to trying to understand functional analysis is quite a thing. I don't like functional analysis and I am not a mathematician either but I have done my share of it. User:D.Lazard or User:JayBeeEll do you want to or do you know any body that can volunteer to fix it ? Flyredeagle (talk) 15:57, 26 December 2020 (UTC)

I have reindented you signature. I hope there is no harm.

I am not an expert of functional analysis. However, I agree that the former lead was better, altough not good. I have not the competence to rewrite the whole lead. However, I suggest the following for the first paragraph.

Functional analysis is an extension of mathematical analysis, which consists to study function spaces and their transformations (which are functionals, that is, functions acting on functions) in view of a finer study of individual functions of real or complex variables. The function spaces that are considered are infinite dimensional topological vector spaces on which are often defined inner products or norms. This point of view allows to use tools of linear algebra and geometry for studying functions.

D.Lazard (talk) 18:14, 26 December 2020 (UTC)

Please review/double check D.Lazard

Flyredeagle (talk) 16:11, 27 December 2020 (UTC)

Please as usual D.Lazard don't throw away, move to talk, review and change for consensus. — Preceding unsigned comment added by Flyredeagle (talk • contribs) 18:12, 27 December 2020 (UTC)

D.Lazard Can you please explain your process to baseline changes, hopefully together .., here ?

Finally from the Historical roots down to Banach is part of history not of definitions, Correct ?

Flyredeagle (talk) 17:33, 27 December 2020 (UTC)

Transformation is a pretty unknown concept where operator is well known and you can't really start functional analysis without operators'

Transformation is defined as automorphism, when you look at duality in banach spaces these are maps they are not automorphisms, therefore using transformations here is wrong.

Mentioning functionals is also a bit misleading still using operators is better

Forgetting metrics, measure and calculus is a piece of total nonsense in this definition

Forgetting generalized functions, distributions, and operators is also nonsense

Last what's wrong with the rest of my contribution ?

Flyredeagle (talk) 17:46, 27 December 2020 (UTC)

I have removed your copy (here) of your version of the lead, because it was unclear that it was a copy of a part of a reverted version of the article. For discussing it, it is better to use this link, which contains the whole removed text (not only the lead), and the reasons of my revert.

My revert was based on several reasons. Firstly, your version included a non-attributed copy of my above suggestion. Secondly this version contains many duplicated links, many links to dab pages and other wrong links. It contains also wrong formulations and dubious assertions. This makes it much worst that MaxEnt's version. I have not a clear opinion between older version and MaxEnt's one. So, further discussion is needed here. I have restored the old version of the lead, as it is usual in case of content disputes. D.Lazard (talk) 17:50, 27 December 2020 (UTC)

I understand the content dispute part

I also accept to revert to original, cause in my page there was the work to be done left over in the page, although I prefer a continuous editing mode.

How you can claim that I reused part of your wording and I shall attribute it some how ? How I attribute to you one word yes and one word not ? There are significant differences in content that the copyright is different actually or better is shared.

Given the licensing of the content the content is perfectly reusable and editable "together". The attribution part how it goes given this is the same paragraph ?

Can you please expand on what are the wrong formulations ? (are you referring to metric ?)

and dubious assertions: Which ones ?

D.Lazard Can you please reply to all the questions including the other questions above ?

Flyredeagle (talk) 18:06, 27 December 2020 (UTC)

Draft new lead

Please review, cross edit, discuss, add and remove Flyredeagle (talk) 11:31, 28 December 2020 (UTC)

General comments and discussion

In my opinion, this draft gives a biased view of functional analysis, and seems WP:Original synthesis. So, it cannot be used as a starting point for improving the article.

The first two paragraphs are essentially my above suggestion, with many linking errors added, and formulation changes that destroy the intended meaning. Nevertheless, these two paragraphs are the only ones that are really related to the subject of the article.

The last paragraph talks of finite dimentional vector spaces (which are not at all part of functional analysis), and considers the basic tools of functional analysis as "advanced examples".

The four other paragraphs contain many other similar conceptual errors. There are too many for being detailed here. Also, all four mention generalized functions, suggesting wrongly that they form the core of functional analysis.

So, in my opinion, this draft is definitely not useful. D.Lazard (talk) 12:58, 28 December 2020 (UTC)

Draft

Functional analysis is an extension of mathematical analysis, which consists to study function spaces and operators defined on such function spaces. This in view of a finer study of individual functions, families of functions and equations of functions of real or complex variables.

The function spaces that are considered are often infinite dimensional topological vector spaces on which there is often a choosen definition of inner product or norm. This point of view allows to use tools of linear algebra and geometry for studying functions.

The choosen definition of norm and the choosen topological structure usually also induce a definition of metric (or heuristically "distance" between functions) and measure and therefore it allows tools from calculus such as the lebesgue integral and the weak derivative to be extended to new types of generalized functions such as distributions and operators.

The generalized functions are then often seen in the context of being solutions of partial differential equations and integral equations defined in a specific function space.

Functional analysis being the study of generalized functions becomes then central in mathematics, in the study of partial differential equations and in the study of operators with extensive applications in mathematics, physics, engineering, statistics and economy. A huge part of functional analysis is also in fact dedicated to the study of linear operators and their Eigenvalues and eigenvectors.

Typical functions are often defined on vector spaces such as $\mathbb {R} ^{n}$ or $\mathbb {C} ^{n}$. Typical simple examples of function spaces are $\mathbf {C} ^{0}(\mathbb {R} )$ (The set of continuous functions on $\mathbb {R}$) or $\mathbf {C} ^{\infty }(\mathbb {R} ^{n})$ (The set of always differentiable functions on $\mathbb {R} ^{n}$). Typical advanced examples of function spaces are instead banach spaces, hilbert spaces and sobolev spaces.

References

And then a section about history in the body reusing the content from the current lead

History The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

The usage of the word functional as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.

References

Flyredeagle (talk) 11:31, 28 December 2020 (UTC)

Extra reusable stuff

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite-dimensional spaces. In contrast, linear algebra deals mostly with finite-dimensional spaces, and does not use topology. An important part of functional analysis is the extension of the theory of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.

Flyredeagle (talk) 11:31, 28 December 2020 (UTC)

Rationales

First Paragraph

• First definition shall be short and sharp
• Removed references to functionals in the first definition: is referred to in the history section and is confusing and distracting.

An operator is a generalized type of functional, that is, functions acting on functions. This can be either a composition of functions, linear, non linear, differential, integral etc.

• individual functions are specific cases of math applications related to things such as the zeta functions etc. This is not the generic setup of functional analysis/
• The core of functional analysis is function spaces i.e. families of functions
• And Functional equations referred as third example
• The current lead confuses linear functional analysis with the overall functional analysis the recent focus is of course non linear and this non linear functional analysis is a first class citizen
• endowing with topology, metric and measure is also added later and separated out, where endowing is replaced with "choosen" . see below.
• The focus of infinite dimensional spaces only is too narrow, it's true that it gives peculiar properties but it is not the general setting: a problem to use a finite set of a function basis instead of the full basis and to use one part of one basis instead to approximate another full basis is crucial for numerical applications. Another example can be spin where the basis is usually finite dimensional.

This paragraph is attributed in a shared manner to D.Lazard with derivative work additions from me (i.e. additions and removals).

Flyredeagle (talk) 11:31, 28 December 2020 (UTC)

Second paragraph

• Topology is moved and references to geometry and linear algebra are here as a second focus

This paragraph is attributed in a shared manner to D.Lazard with derivative work additions from me (i.e. additions and removals)

• I addded "choosen" topology and choosen norm/product on purpose, given you can have different combinations of topologies, norm, product, measure, metrics, integral definitions and derivative definitions that gives a final set of properties that can be different. Namely finite topologies are perfectly legitimate and same is valid for sesquilinear forms just to name one.

Flyredeagle (talk) 11:31, 28 December 2020 (UTC)

Third paragraph

• Measure theory, metrics and weak derivatives are moved here, I have choosen weak derivative given on it's own can drive a definition of the rest. I know is wobbly but is the best to show that we are generalizing calculus and analysis (i.e. both the integral and the derivative).
• the heuristic example "distance" between functions is still borrowed from function fitting, with respect to different norms, it can be eventually removed if it is too fancy

Flyredeagle (talk) 11:31, 28 December 2020 (UTC)

fourth paragraph

Added the relationship with PDEs and integral eqts. to show where is the central focus

Flyredeagle (talk) 11:31, 28 December 2020 (UTC)

fifth paragraph

Added another big section for applications and centrality in math, i.e. the typical "pure math" drama.

Flyredeagle (talk) 11:31, 28 December 2020 (UTC)

Six paragraph

Addded another section with examples of function spaces to harden it to the ground.

Flyredeagle (talk) 11:31, 28 December 2020 (UTC)

History section

• history does not belong to definition and lead
• the history is long enough to have it's own space so that it can be expanded further
• current quality of history section is good for the main body
• functional is defined here and is good enough a possible addition can be that operators are generalized functionals.

Flyredeagle (talk) 11:31, 28 December 2020 (UTC)

Removals

• Transformation is defined as automorphism, when you look at duality in banach spaces these are maps they are not automorphisms, therefore using transformations here is wrong.
• Linear is removed given the full scope is non linear, reference to it is left as a big chunck is Linear ...
• I removed "limit related structure" given it fits more and is more precisely defined, in the scope of topology, linear infinite dimensional spaces, metrics, measure etc. It's a nice to have but unclear how to fit the text (i.e. given sharpness of first section).

Flyredeagle (talk) 11:31, 28 December 2020 (UTC)

Additions

• Generalized functions - here the context is that we are also generalizing functions to "other stuff"
• distributions - Schwartz
• operators: subject of functional analysis are also operator equations, which e.g. in the banach case they sit on the dual space. e.g. Heisenberg equations. Von neumann theorems etc.

Flyredeagle (talk) 11:31, 28 December 2020 (UTC)

Other improvements TBD

• check links
• add more refs
• please review text and correct it (I am a non native speaker)

Flyredeagle (talk) 11:31, 28 December 2020 (UTC)