Talk:Functional analysis

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Mathematics rating:	Start Class	Top Priority	Field: Analysis
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. 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)

functionnal analysis starts with fréchet but you don't say it —Preceding unsigned comment added by 90.46.26.92 (talk) 21:04, 21 December 2007 (UTC)

[edit] 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 21:27, 11 March 2007 (UTC)Maria

[edit] Merge?

The following discussion relates to a now abandoned proposal to merge this article with calculus of variations.

Really, no way should this page be merged with calculus of variations. Phys, my man, these subjects diverged nearly a century ago. Charles Matthews 07:56, 5 Sep 2003 (UTC)

I'm inclined to agree here. Dysprosia 08:01, 5 Sep 2003 (UTC)

Additional remarks. I'm sure much of this is wrong, and much of the rest is POV. Some infinite-dimensional spaces can be shown to have a basis without AC, surely. Hahn-Banach can be side-stepped in separable spaces. And the advice is not necessarily good. Is any of this really important to keep?

I fixed the remark saying that Hahn-Banach requires AC. Hahn-Banach can only be side-stepped in separable spaces if one allows DC. The comment about infinite-dimensional spaces is correct, I would think, because almost all spaces actually considered in functional analysis can not be shown to have a basis without essentially using AC for some cardinal. However, there is no reason why it should say Zorn's lemma instead of the axiom of choice, and the point is largely irrelevant for functional analysis, as there are very few cases where one actually needs a Hamel basis for a particular infinite-dimensional Banach space. Cwzwarich 02:56, 4 December 2005 (UTC)

Charles Matthews 19:38, 8 Sep 2004 (UTC)

"Since finite-dimensional Hilbert spaces are fully understood in linear algebra (...)"

That's not completely true. In fact, very much is unknown even about the geometry of !

mbork 15:16, 2004 Nov 23 (UTC)

Is it really kind to give the three volumes of Dunford and Schwartz as basic reference? Charles Matthews 18:42, 23 Nov 2004 (UTC)

[edit] Hahn-Banach theorem

The error was saying that the Hahn-Banach theorem requires the axiom of choice when it doesn't. Cwzwarich 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 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 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 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)

[edit] I don't understand this

In the article it was said that "every operator on a Hilbert space has a proper subspace which is invariant". Now is invariant some functional theoretic term or is that invariant under some transformation? I'm not an expert on FA. --Matikkapoika 19:31, 18 February 2006 (UTC)

I fixed that, I believe it is about invariant subspace. By the way, you should not use the minor edit checkmark for edits which are not minor, like for example starting a new topic on talk pages. Also, it is good to use edit summaries, so that others can tell what you changed. Thanks. Oleg Alexandrov (talk) 00:05, 19 February 2006 (UTC)

[edit] ================

I have a few comments about wording and substance of a few claims raised in this article:

1."General Banach spaces are more complicated." (in "Banach spaces" paragraph) I assume that this relates Hilber and Banach spaces. This is a type of "apples or bananas" issue and there is no need to try to resolve it here.

2. "There is no clear definition of what would constitute a base, for example" (very next sentence). This is definetely not true. The notion of a base in Banach spaces is well defined and understood, see any text book on Functional Analysis. Moreover, this sentence seems to relate somehow to its predicessor and suggests that Banch spaces are so difficult to understand that people do not even know how to define a base.

3. "For any real number p ≥ 1, an example of a Banach space is given by "all Lebesgue-measurable functions whose absolute value's p-th power has finite integral" (see Lp spaces)" ("Banch spaces" paragraph) Since this sentence is supposed to present an example of Banch space for a general audience, I would suggest that l^p (or C[0,1]) be used instead of L^p. I mean, one would not have to invoke concept of Lebesgue mesaurable functions.

4."To show the existence of a vector space basis for such spaces may require Zorn's lemma"

(Second sentence in "Foundation of mathematics" paragraph)

I am unaware of a singel case of a Banch space which requires use of Zorn's lemma to show that it has a basis.Usually, a basic sequence is "more or less a natural choice" and the prove is focused on showing that, indeed, the sequence forms a basis.

5."geometry of Banach spaces, a combinatorial approach as in the work of Jean Bourgain" (in "Points of view"). The study of geometry of Banach spaces requires much more than combinatorics. Also, the study was initiated by Fritz-Johnes and Dvoretzky work and then developed into "theory" by works of V.Milman (and others). I am not trying to take anything from outstanding work of J. Bourgain, I am just saying that this part of the article contains two misleading references.

6. References section. It contains references to books at least 50 years old. I suggest that the following be added:

1. "Classical Banach spaces" by Joram Lindenstrauss, Lior Tzafriri, ISBN: 3540606289, Springer 1996 2."The Volume of Convex Bodies and Banach Space Geometry " by by Gilles Pisier,ISBN: 0521364655 ,Cambridge University Press

[edit] Functional Analysis is also an important part of Applied Behavior Analysis

Functional Analysis is also a well known term in Applied Behavior Analysis (a branch of psychology) and concerns the analysis of the function of a particular behavior, especially problem or maladaptive behaviors. I would suggest a cross reference to Applied Behavior Analysis.

[edit] ISBN for Analyse Fonctionnelle by H. Brezis

There are two versions, with ISBNs 978-2100493364 and 978-2100043149. Unfortunately, none of the book finder links in wikipedia's book finder seem to work, and it's not on amazon.com, but both versions may be found at amazon.ca, and the former may be found at amazon.fr. Go figure. However, I still think it's worth listing. Althai 17:46, 26 March 2007 (UTC)

[edit] 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 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 03:15, 16 May 2007 (UTC)