Talk:Hahn–Banach theorem

message: I'd like to rename this page to "Hahn-Banach theorem" (with an ordinary hyphen instead of the Unicode dash) and fix the links and get rid of the redirect.  Any reason not to do that?  It took a lot of head scratching for me to figure out how "Hahn-Banach theorem" could be redirected here, i.e. that the title of this page is actually something different. 70.231.131.185 05:25, 9 February 2006 (UTC)
replies: text: Please don't do things like that. There is no reason to do it. It's just a time-sink going over this with everyone, but there is a good copy editor's convention that is being followed here. Charles Matthews 21:53, 9 February 2006 (UTC)
replies: text: Blecch.  The reason to do it is because the hyphen is much easier to type, is almost always what people type, and is what almost all the internal wikipedia links use, which means that with the en dash in the article title, all those links go to redirects and sometimes double redirects.  The reason not to do it is in WP:MOSDASH which I just found.  Maybe the solution is to adopt a convention of writing a double hyphen for the en dash, like in TeX, and put a code patch into the MediaWiki software to transform the double hyphen into the Unicode symbol, sort of like TeX does.  Triple hyphen would similarly transform to an em dash.  I can work on that patch if it's worthwhile, but I probably won't get a chance to do it real soon (there's other stuff I need to finish first).    It would also have to get wider discussion since it would affect a lot of places where people typed double hyphens to express something like an em dash.  Let me know what you think.  (Actually I see now there's a huge discussion on WP:MOSDASH's talk page about the double hyphen suggestion.  In my not-so-copious free time I'll someday try to get around to finding out what happened with it. Phr 00:41, 11 February 2006 (UTC)
thread_title:  redirect? 

message: the sub-linear function defined N:V->R
but isn't it possible for it to have infinite values too? --itaj 00:23, 3 May 2006 (UTC)
replies: text: I think that's a valid point.  The book I looked at specifically mentions that N must be a finite sub-linear functional, since sub-linear functionals can be infinite-valued. Lavaka 18:30, 22 August 2006 (UTC)
thread_title:  infinite sub-linear function 

message: The statement of the H-B theorem says the range of N is the reals, while the range of \psi and \phi are the scalar field K.  This seems odd to me.  We require that the extension \psi be dominated by N, but how to compare the values of the functions if one is in the reals and the other in K?  Earlier, it is mentioned that K is the scalar field (either the reals or complex).  This seems sloppy: do we restrict K to just the reals and complex, or was the parenthetical comment just meant as an example, in which case it should use a word like "e.g." or "ex.".  If it is restricted to the reals or complex, then we can compare N and K using the absolute value or complex modulus, but this should be made clear.   The book I use simplifies everything and assumes K is the reals and doesn't get itself into trouble. --Lavaka 18:48, 22 August 2006 (UTC)
thread_title:  fields 

message: At least on my computer screen, the mathematical formulae used in the article text are barely readable, and I would like to translate them to TeX, unless anybody disagress. — Tobias Bergemann 11:40, 4 May 2007 (UTC)
replies: text: Translating inline formulas to TeX may cause some of them to become PNGs, and one some screens those look huge. The math style manual kind of discourages that. As long as you attempt to minimize inline PNGs I guess translating to TeX should be fine. By the way, the html formulas look perfect on my own screen. Oleg Alexandrov (talk) 15:28, 4 May 2007 (UTC)
thread_title:  TeXify math? 

message: The story of Eduard Helly proving Hahn-Banach
20 years before Hahn and Banach seems to be supported
by Mac Tutor .
I suppose we should add it. -- Tiphareth (talk) 11:45, 19 November 2007 (UTC)
thread_title:  Eduard Helly 

message: Isn't the formulation for convex functions as seen in Reed and Simon the most general for R vector spaces? I think this should be added somewhere. Also it should be made clear that the other formulation from Reed and Simon, the last in the section "Formulation" is meant for complex vector spaces. Quiet photon (talk) 18:40, 8 May 2010 (UTC)
thread_title:  Most general formulation 

message: Apart from "Consequences" which are direct corollaries couldn't any non-trivial applications be mentioned?  —Preceding unsigned comment added by 78.157.76.143 (talk) 18:33, 23 May 2010 (UTC)
thread_title:  Applications 

message: I think the main theorem (the first stated version of the Hahn-Banach theorem) with the bound |f| \le p, where p is a sublinear function, is false.
A counterexample: Let \mathbb{K} = \mathbb{R}, X = \mathbb{R}^2, p(x,y) = |x|-x+|y|-y, M = \mathbb{R}(1,-1), f(x,-x) = 2x.
Then p is a nonnegative sublinear function (but it is not a seminorm), and |f| = p on M, however there is no linear extension F of f to X such that |F| \le p, because p(1,0) = p(0,1) = 0, so F would have to be constant 0, but f \neq 0.
The theorem is true if p is a seminorm, see Theorem 3.3 in Rudin.
The theorem is also true for sublinear p, if we omit the absolute values, so we use f \le p on M and F \le p on V, see Theorem 3.2 in Rudin.
188.36.117.186 (talk) 17:05, 23 June 2021 (UTC)
thread_title:  False main result with sublinear functions instead of seminorms 

message: The subsection "Partial differential equations" needs, at the least, a reference. Moreover it seems to me that it has some content problems. It is not correct that the adjoint of a linear operator is defined only on the image of the operator. And it does not seem to make sense that a standard solution () is being extended by Hahn-Banach to a weak solution; it is already a standard solution and hence is itself a weak solution by any reasonable definition. Gumshoe2 (talk) 17:35, 8 September 2022 (UTC)
thread_title:  Problematic PDE section 

message: The redirect Mazur's theorem has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached.   Jay 💬 13:15, 13 February 2024 (UTC)
thread_title:  "[[:Mazur's theorem]]" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]]