Talk:Banach space

message: The TOC link for section 2.8.1 isn't rendering correctly - shows up as "2.8.1	Results involving the '"`UNIQ--postMath-00000289-QINU`"' basis.   — Preceding unsigned comment added by Alexmay28 (talk • contribs) 16:37, 28 June 2022 (UTC)
thread_title: TOC link displaying incorrectly

message: As a non expert reader, I have no idea what the use of the concept of a Banach space is. Does it help in understanding elementary particles, cryptography or computer simulation of physical problems? Is the concept of a Banach space useful to prove other critical mathematical theorems?
The utility may not be the reason that mathematicians are inspired to work on Banach spaces. Any useful purposes are still helpful to most readers though to help them understand why Banach spaces are important.
I've included a first attempt at saying something about this to generate some discussion.  — Preceding unsigned comment added by Chogg (talk • contribs) 11:37, 6 September 2012 (UTC)
replies: text: Banach spaces play a vital role in the mathematical analysis of everything from relativity to materials science.  Probably part of the reason there is no "applications" section is that Banach spaces are almost too ubiquitous.  I'm not saying that there shouldn't be one, just that the task of doing so seems to require an almost encyclopedic knowledge of modern analysis.  Sławomir Biały (talk) 12:47, 6 September 2012 (UTC)
thread_title:  Applications 

message: Do any of the examples have commonly used names, such as the "blah" space or the "blah" Banach space? For instance, I have in front of me the "Banach space of all analytic functions on a domain D with sup norm", and am fishing for a shorter term for this. Also: symbology: the text in front of me calls this space A_\infty(D) ... Is this a common notation for this? Why infty? I presume there is a "Banach space of polynomials of degree p on domain D with sup norm" called A_p(D) ? (Dohh its the sup norm, silly me.) linas 14:05, 19 September 2005 (UTC)
thread_title:  Common names and notation for examples? 

message: In the opening paragraph, should there be a comment that Banach spaces may be finite dimensional as well as infinite dimensional?   ~ Moocowpong1 00:20, 7 November 2005 (UTC)
replies: text: I like the way things are now. It says "typically infinite-dimensional" which does not imply "only infinite-dimensional". And indeed, the truly useful case of Banach spaces is in the infinite-dimensional case. So I would leave it the way it is. Oleg Alexandrov (talk) 02:26, 7 November 2005 (UTC)
thread_title:  Infinite dimensional? 

message: linas, it seems that you reverted some anon who removed this stuff a section saying that the trace yields an isomorphism between a space and its dual.  I can't understand how this section, as it currently stands, the way you've reverted it, can be correct, so perhaps you can explain it to me?
So we have a map F: V*xV → K.  This map can be called the trace map (it is the trace map in a finite dim space).  By currying, you propose to give us a map V → V*.  But it seems to me that currying here can only give you an element of V**.  And this is borne out by the formula, because F(-,x) seems to be an element of V**: it takes an element of V* like f, and returns a scalar F(f,x)=f(x).  In short, I think what you've got here is a longwinded notation to describe the well known injection V → V** given by f ↦ f(x). Have I made a mistake? -lethe talk 18:00, 22 November 2005 (UTC)
replies: text: Nope, sorry, I made the mistake. The original version so breif, I thought it was trying to curry. I tried to make that clear, and as you point out, I didn't do it right. The current version looks good. linas 20:13, 22 November 2005 (UTC)
replies: text: Actually, now that I think about it, I completely bloopered it, didn't I? I sat there trying to get te wording right, and it kept coming out wrong. That alone should have been a warning sign ... linas 20:15, 22 November 2005 (UTC)
thread_title:  linas' recent revert 

message: I see that recently, Oleg, that you changed V' to V* and V to V** in order to "make it agree with Dual space".  If you read that article, you will see that both V' and V* are used, but to mean different things; the first is the continuous dual, and the second is the algebraic dual.  I'm not sure whether this is standard.  Seems to me that the dual space should always be a construction within the category, and so the continuous dual is the important one.  Anyway, whatever notation we use we should make it clear what we're talking about.  I didn't revert your edit, however, because I think the injection V → V** is true for both continuous duals and algebraic duals, and the latter subsume the former, being more general, and so your notation agrees with the most general statement.  I just want to make sure that you were aware of the implications of your edit.  If you were, forgive my presumption. -lethe talk 18:24, 22 November 2005 (UTC)
replies: text: You are right. I was aware of the distinction between dual space and continuous dual space, but not about which notation is used for which one. There is no problem with Banach space itself, as there the * is used consistently to mean the continous dual. But I will now change it back to prime to be indeed as in dual space.  Oleg Alexandrov (talk) 01:53, 23 November 2005 (UTC)
replies: text: The best thing to do when there is an ambiguity is to just explain it.  I've added my attempt at explanation. -lethe talk 11:55, 19 December 2005 (UTC)
replies: text:  Thanks! Oleg Alexandrov (talk) 16:13, 19 December 2005 (UTC)
thread_title:  changing V' to V* 

message: This is a very fine, easy to read, well-written article. Most of the credit goes to AxelBolt I would say. Great job. Oleg Alexandrov (talk) 00:26, 13 December 2005 (UTC)
thread_title:  Nice article 

message: I quote the article:
"Since the norm induces a topology on the vector space, a Banach space provides an example of a topological vector space..."
I don't understand why the norm induces a topology... I would say that a norm induces a metric...
It's just an opinion from a guy that doesn't understands much of this, so I leave it to someone else.
--201.253.45.251
:The norm induces a metric, which induces a topology. So it's OK. --Zundark 19:06, 8 October 2006 (UTC)
thread_title:  Possible correction 

message: Small complaint about the section on Hilbert spaces, in the sketch of the proof that parallelogram identity implies inner-product norm.  The article says:
"the completeness of the norm extends the linearity to the whole real line" from the rationals.  However, it is actually the continuity of the norm that is being used here.  Suppose ax is in the space.  Take a sequence of rationals q_i converging to a, and regardless of completeness q_ix -> ax.  The continuity of (. , .) finishes the proof. 84.0.252.215 10:43, 28 February 2007 (UTC)
thread_title:  Possible correction (take two) 

message: Should there be a little bit about banach manifolds in this article? I would think yes but perhaps that deserves its own separate content.
Topology Expert (talk) 04:33, 10 October 2008 (UTC)
thread_title:  [[Banach manifold]]s 

message: I've a question about sequence spaces which most books (and this article) seem to shy away from, and its confusing me, yet it seems central to Banach spaces.  The space of all real-valued sequences with sup norm, i.e. the space of all f:\mathbb{N}\to\mathbb{R} with sup norm, is not a Banach space (at least not normally, not that I'm aware of), right? But why? I know that the "classical" answer would be "because N is not compact", or that by simply limiting one-self to bounded sequences, one would get a Banach space.  That's the classical approach.  
But the modern definition seems to be "complete normed vector space".  Now, perhaps I'm being really dumb here, but the space of all real-valued sequences meets this definition: its a vector space, its normed, and its complete.  Now, this space is "sick", in the sense that many of its elements have infinite norm (e.g. the sequence a_n=n has infinite norm), but the "modern" definition doesn't rule out infinite norm, right? The word "bounded" wasn't in the definition.  The space is also "sick" in that all sequences of infinite norm appear to be topologically indistinguishable -- so e.g. a_n is not distinguishable from b_n=n^2 by the norm ... but so what? The "modern" definition does not require the Banach space to be separable, Hausdorff or T_1 or T_0 or anything. So what gives? What am I missing? linas (talk) 00:30, 9 November 2008 (UTC)
thread_title:  Sequence space 

message: 'Banak' does not make any sense, see Stefan Banach.  —Preceding unsigned comment added by 212.87.13.78 (talk) 20:21, 7 March 2010 (UTC)
thread_title:  Pronunciation 

message: The plus sign in the polarization identity is right provided that ⟨x,y⟩ is linear in its first argument. "In some conventions, inner products are linear in their second arguments instead" (quote from Hilbert space), in which case the minus sign in the polarization identity is right. Boris Tsirelson (talk) 18:59, 31 August 2011 (UTC)
replies: text: ah, in this case you are right. I added a small info above this formula. I am working with linearity in second argument and it didn't cross my mind that it could be otherwise here, so I think that this information should be close to this formula - but please feel free to rephrase it. Janek Kozicki (talk) 07:05, 1 September 2011 (UTC)
thread_title:  "Small factual correction" 

message: "A piece of paper is a complete vector space, unless you omit the points exactly on the edge." — No; it is rather a complete metric space. A bounded part of a vector space cannot be a vector space (unless it is the origin only). Boris Tsirelson (talk) 04:52, 17 June 2012 (UTC)
thread_title:  "less technical intro paragraph" added 

message: Thanks goes to editor Bdmy for a initiating a nice reorganization and cleanup of this article. Regarding cleanup, I noticed that the same (or nearly the same) large table of classical Banach spaces is present in both the Examples section of this article and in the List of Banach spaces article. We probably don't need this duplication. Would it be better to delete it from this article, or perhaps keep just a few examples? Or would it be better to fold back in the content from List of Banach spaces? --Mark viking (talk) 20:06, 28 April 2013 (UTC)
replies: text: Thanks for your thanks! I believe that one table is enough, and then, it is probably better and more logical to have it in List of Banach spaces, with just a link from here to there, and only some examples in the text of this article. Bdmy (talk) 20:39, 28 April 2013 (UTC)

text: I think it should be kept for now, but ultimately it would be better to give a few examples in greater detail for illustration purposes with a link to the main list.  Sławomir Biały (talk) 12:04, 17 June 2013 (UTC)
thread_title:  Duplicated list of classical Banach spaces 

message: I do not usually totally erase contributions that other made the effort to write, so I think I have to explain two suppressions I made :
:* The definition of Banach space had an inline citation attached to it, although it is just the usual definition, as it can be found in the Wikipedia article Complete metric space. Also, the choice of the book seemed a bit strange to me.
:* Math Reviews says it has registered over 600.000 author names, but "Willkomm, Anton" is unknown there. I don't think that "Willkomm, Anton (1976), Über die Darstellungstheorie topologischer Gruppen in nicht-archimedischen Banach-Räumen, Dissertation, Rheinisch-Wetfälische Technische Hochschule Aachen." is a notable reference for Wikipedia, and I removed it from "References".
I feel that I finished (most of) what I wanted to do in order to meet the remarks in this Talk pages by Mct mht and Sławomir Biały. Since English is not my mother tongue, it could be useful that someone with English expertise read and correct my awkwardness.
I may made further additions later, for example to (shortly) review some of Banach space theory since around 1980. This should take more time than just reorganize.
Bdmy (talk) 21:28, 1 May 2013 (UTC)
replies: text: Approximation problem redirects to compact operators. Approximation property is probably a better target (although that page doesn't say what the problem is either, presumably about whether CAP the identity map can be approximated by finite rank operators?). Mct mht (talk) 20:23, 2 May 2013 (UTC)

text: Oops, should've read the intro. Approximation property still seems more appropriate, though. Mct mht (talk) 20:39, 2 May 2013 (UTC)
replies: text: I changed the link. Also, I'll have to improve my vague statement about tensor products, Grothendieck and the approximation property. Bdmy (talk) 09:50, 3 May 2013 (UTC)
thread_title:  about my recent "General cleaning" 

message: I propose to remove the figure at Banach space#Placement in the hierarchy of mathematical structures.  In my opinion, it is not very informative.  But more than this, there is no particular reason to have it here rather than at "Hilbert space", "metric space", "topological space",...  If nobody objects, I'll remove it in about a week. Bdmy (talk) 18:17, 4 June 2013 (UTC)
replies: text: I like the illustration, but agree that doesn't really belong here. Perhaps someday an infobox of mathematical spaces will be created that accomplishes the same purpose of placing Banach spaces in a greater context. In the meantime, I have copied the illustration and caption over to Space (mathematics)#Types of spaces, which does have a nice comparative overview of mathematical spaces and could use the illustration. I have no objections to removal. --Mark viking (talk) 21:04, 4 June 2013 (UTC)
thread_title:  Placement in the hierarchy of mathematical structures 

message: I plan to get rid of the "Further results" section, where I have moved far down, some time ago, a bunch of randomly picked results with no context.  These results were part of the "long bulleted list" that appeared in this article around September 2012.  If noone objects, I'll remove it in about a week.
:I did it. Bdmy (talk) 12:30, 13 July 2013 (UTC)
replies: text: Help didn't come! (yet) Bdmy (talk) 12:30, 13 July 2013 (UTC)
thread_title:  Next cleaning 

message: Excuse me, in which sense for example ℓ1 is weakly complete? Eozhik (talk) 09:19, 26 July 2013 (UTC)
replies: text: The table has been here for a long time, and also appears at List of Banach spaces.  I think it comes from Dunford and Schwartz.  Clearly the meaning is weakly sequentially complete (see in the text, at Banach space#Weak convergences of sequences).  There are also a few errors in the table, where duals isomorphic to ℓ∞ are marked as being ℓ1.  I had the plan to clean this some day, but it is not my highest priority (I don't appreciate this table very much.  I suggested to remove it, since it appears elsewhere, but some opposed and I respected their reasons). Bdmy (talk) 09:32, 26 July 2013 (UTC)
replies: text: Ah, I see... However, somebody must put "sequentially" into this place of the table (and correct the links), because otherwise this will be a source of misunderstandings. Eozhik (talk) 12:25, 26 July 2013 (UTC)
thread_title:  What does weak completeness mean in this table?  

message: "The unit ball of the dual is ... metrizable for the weak* topology" — Really? Let us consider a nonseparable Hilbert space. The weak* (=weak) topology of its ball has no countable base at the origin. Moreover, the intersection of countably many neighborhoods of the origin is never the origin only; it always contains a nonseparable subspace (intersected with the ball). Right? Boris Tsirelson (talk) 06:00, 6 May 2014 (UTC)
thread_title:  2.8.2 Sequences, weak and weak* compactness 

message: "If it is known that the projection operators  are continuous, then" — No, we do not need to assume this continuity; we can prove it. See for example Matthew Daws, Theorem 2.1. Boris Tsirelson (talk) 21:20, 22 October 2014 (UTC)
thread_title:  Shauder basis 

message: The first link in the reference section is dead:  Théorie des opérations linéaires http://kielich.amu.edu.pl/Stefan_Banach/pdf/teoria-operacji-fr/banach-teorie-des-operations-lineaires.pdf  — Preceding unsigned comment added by Bongilles (talk • contribs) 08:31, 31 January 2018 (UTC)
thread_title:  Dead link 

message: In the footnote linking to Szankowski's paper proving that B(H) does not have the approximation property, there is a remark stating (correctly) that Ryan's book on tensor products attributes this to Enflo. I think this is a confusion by Ryan -- Enflo first constructed a Banach space without the approximation property, but it was Szankowski who found the first proof that the approximation property did not hold for a well-known space, not specially constructed for the purpose (i.e. B(H)). Certainly I cannot find any paper by Enflo on whether or not B(H) has the approximation property, nor any other attribution of this result to Enflo.
What is the established procedure for such things? Clearly we should not state in the article that it is correct that Enflo proved B(H) does not have the approximation property -- this is explicitly contradicted in the introduction to Szankowski's article. But as it is someone might come along later and blithely use that remark in Ryan's book as a source. RobertFurber (talk) 16:25, 5 June 2022 (UTC)
thread_title:  Attribution of B(H) lacking the approximation property to Enflo 

message: I don't understand the pronunciation guide given for 'Banach space'; 'Banach space' is 3 syllables, but the number of syllables in the pronunciation guide is 2. If the pronunciation guide is only for the word 'Banach', which I guess seems likely, as we don't need a guide for how to pronounce 'space', I still don't understand it, because it seems like that 'x' should be a 'k'. Kontribuanto (talk) 12:08, 23 February 2024 (UTC)
thread_title:  possibly incorrect pronunciation guide given