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755:: "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
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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
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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
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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
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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
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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,
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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
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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
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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
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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
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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
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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
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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.
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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.
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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
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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.
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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
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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
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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.
887:]. 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.
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669:"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".
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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?
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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".
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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
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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.
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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.
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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.
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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
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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.
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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/
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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 ?
663:"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."
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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.
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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
1694:'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
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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.
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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.
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deals mostly with finite-dimensional spaces, and does not use topology. An important part of functional analysis is the extension of the theory of
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So as not to leave this article in different persistent state of neglect, I aggressively flagged my guesswork with expert-needed and speculation.
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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:
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In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular
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I have an intermediate level of mathematical sophistication, having once completed half a math degree, and having worked in STEM thereafter.
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The current lead confuses linear functional analysis with the overall functional analysis the recent focus is of course non linear and this
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The four other paragraphs contain many other similar conceptual errors. There are too many for being detailed here. Also, all four mention
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I have removed the link added with the summary "See also: link to Anna
Johnson Pell Wheeler (famous for theories on func. analysis". From
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Soft analysis redirects here, but nothing explains why or how this is considered "soft". The phrase itself is never used in the article.--
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Transformation is a pretty unknown concept where operator is well known and you can't really start functional analysis without operators'
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endowing with topology, metric and measure is also added later and separated out, where endowing is replaced with "choosen" . see below.
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Functional analysis (sometimes "functional harmonic analysis" or "harmonic analysis") in music theory refers to the formal analysis of
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This paragraph is attributed in a shared manner to D.Lazard with derivative work additions from me (i.e. additions and removals).
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Removed references to functionals in the first definition: is referred to in the history section and is confusing and distracting.
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This paragraph is attributed in a shared manner to D.Lazard with derivative work additions from me (i.e. additions and removals)
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My agenda here was simply to learn the material, and I made these edits passing through. Consider the torch passed. —
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functional is defined here and is good enough a possible addition can be that operators are generalized functionals.
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Linear is removed given the full scope is non linear, reference to it is left as a big chunck is Linear ...
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Notes in Math. 369, Springer, 203-248. Thanks for letting me know about the other page. I will correct it.
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Finally from the
Historical roots down to Banach is part of history not of definitions, Correct ?
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The error was saying that the Hahn-Banach theorem requires the axiom of choice when it doesn't.
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Can you please expand on what are the wrong formulations ? (are you referring to metric ?)
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is another unrelated concept in mathematics that's sometimes used to mean the same thing.
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I don't know which is true, but to me it looks like there is some kind of contradiction.
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Forgetting metrics, measure and calculus is a piece of total nonsense in this definition
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Added the relationship with PDEs and integral eqts. to show where is the central focus
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And then a section about history in the body reusing the content from the current lead
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Can you please explain your process to baseline changes, hopefully together .., here ?
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the history is long enough to have it's own space so that it can be expanded further
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Addded another section with examples of function spaces to harden it to the ground.
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In the "Foundations of mathematics considerations" section, there is a statement:
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they say that it requires Zorn's lemma, which is the same as the axiom of choice.
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It's an imprecise slang term, and the redirection reflects only one of its uses.
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In my opinion, this draft gives a biased view of functional analysis, and seems
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perfectly legitimate and same is valid for sesquilinear forms just to name one.
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Can you please reply to all the questions including the other questions above ?
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Forgetting generalized functions, distributions, and operators is also nonsense
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Mentioning functionals is also a bit misleading still using operators is better
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defined on these spaces and respecting these structures in a suitable sense.
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and the formulation of properties of transformations of functions such as the
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acting on functions) in view of a finer study of individual functions of
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Axiom of choice is strictly weaker than the
Boolean prime ideal theorem ?
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The function spaces that are considered are often infinite dimensional
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do you want to or do you know any body that can volunteer to fix it ?
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Functional analysis includes both some soft and some hard analysis.
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is an extension of mathematical analysis, which consists to study
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don't throw away, move to talk, review and change for consensus.
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The historical roots of functional analysis lie in the study of
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please review text and correct it (I am a non native speaker)
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current quality of history section is good for the main body
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there is no difference whether you use choice or the BPI).
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I have reindented you signature. I hope there is no harm.
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are then often seen in the context of being solutions of
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endowed with some kind of limit-related structure (e.g.
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So, in my opinion, this draft is definitely not useful.
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textbooks, the underlying mathematics remains the same.
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Last what's wrong with the rest of my contribution ?
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2160:Flyredeagle
2149:check links
2131:Flyredeagle
2102:Flyredeagle
2072:Flyredeagle
2040:Flyredeagle
2019:Flyredeagle
1998:Flyredeagle
1977:Flyredeagle
1950:Flyredeagle
1918:Flyredeagle
1882:, that is,
1848:Flyredeagle
1837:probability
1833:integration
1769:Flyredeagle
1373:engineering
1365:mathematics
1353:mathematics
1182:Flyredeagle
1156:Flyredeagle
1108:worst that
1088:Flyredeagle
1054:Flyredeagle
1024:Flyredeagle
994:Flyredeagle
938:, that is,
936:functionals
934:(which are
897:Flyredeagle
801:—Preceding
348:Mathematics
339:mathematics
295:Mathematics
161:free images
44:not a forum
2176:Categories
1880:functional
1861:Rationales
1727:References
1679:functional
1659:continuous
1615:References
1387:and their
1377:statistics
930:and their
915:paragraph.
709:Prosfilaes
561:80.6.80.51
2115:Additions
2097:section).
1884:functions
1361:operators
1323:operators
1280:functions
1249:operators
1102:this link
940:functions
753:Mac Tutor
539:Cwzwarich
505:Cwzwarich
264:is rated
101:if needed
84:Be polite
34:talk page
2085:Removals
1808:topology
1710:and the
1692:Hadamard
1627:Conway.
1398:such as
1317:such as
1309:and the
1303:calculus
1276:geometry
1222:D.Lazard
1150:D.Lazard
1118:D.Lazard
1042:D.Lazard
1032:contribs
1020:unsigned
1014:D.Lazard
989:D.Lazard
975:D.Lazard
968:geometry
844:shotwell
803:unsigned
781:Jowa fan
761:Jmath666
674:Alkelele
440:365 days
406:Archives
212:365 days
199:Archives
69:get help
42:This is
40:article.
1829:measure
1700:Fréchet
1663:unitary
1647:History
1381:economy
1369:physics
1299:measure
1137:shared.
948:complex
759:, etc.
375:on the
266:C-class
167:WP refs
155:scholar
1835:, and
1760:p. 195
1716:Polish
1295:metric
1114:MaxEnt
1110:MaxEnt
877:MaxEnt
689:John Z
485:Ormaaj
272:scale.
139:Google
1712:group
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1522:) or
1235:Draft
960:norms
638:Temur
597:Temur
253:This
182:JSTOR
143:books
97:Seek
2164:talk
2135:talk
2106:talk
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2044:talk
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2002:talk
1981:talk
1954:talk
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1804:norm
1773:talk
1755:ISBN
1704:Lévy
1702:and
1669:and
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1460:are
1379:and
1336:and
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998:talk
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175:FENS
149:news
86:and
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