Talk:Functional analysis
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| Mathematics grading: | Start Class | Top Importance | Field: Analysis | ||
| This article doesn't do enough to emphasis the incredible importance of functional analsysis. 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 08:35, 8 October 2006 (UTC) | |||||
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[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)
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- 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)
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[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)
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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)
