Talk:Hahn–Banach theorem

Generalizations and Counter-Examples

In addition to failing in non-convex topological vector spaces for the domain, Hahn-Banach fails for maps into normed vector spaces. It would be nice to point out some results in this direction.

mistake?

In the second important consequence, in the last formula instead of 1/||z|| must be 1/dist(z,U). The same mistake in other languages except for Polish.Dy76 (talk) 13:27, 19 January 2008 (UTC)

if V is a normed vector space with subspace U (not necessarily closed) and if z is an element of V not in the closure of U, then there exists a continuous linear map ψ : V -> K with ψ(x) = 0 for all x in U, ψ(z) = 1, and ||ψ|| = ||z||-1.

I think "not necessarily closed" is wrong here. But can't make up a counterexample right now.

What do you mean "wrong" ? It's a hypothesis, it cannot be right or wrong... It could be unnecessary to mention it, but it can't be wrong.

Uhm, he means that the assertion is false, when (not necessarily closed) appear where it does. The implication is rendered false.

However, it is seems ok as it stands, since z is assumed to be outside the closure of U, and thus have positive distance to U.

But, as formulated above it is surely false. There will be small elements close to any subspace, and thus the norm of the functional would be negative... remove the -1 and it seems fine.

AC, Hahn-Banach and Banach-Tarski

Suppose you assume ZF as the axioms of set theory (i.e. without AC or its negation). You also assume the Hahn-Banach theorem is true. Can you prove AC? If yes, I think that means you can't get rid of the Banach-Tarski paradox without also losing the Hahn-Banach theorem. Is that interesting? 70.231.131.185 18:37, 8 February 2006 (UTC)

The following implications hold:

AC → BPI → "BM" → Hahn-Banach theorem → Banach-Tarski paradox.

where "BM" means that on every Boolean algebra there is a finitely additive measure.

1. The first implication is of course well-known, and it is known that the reverse does not hold.
2. The second implication is trivial (take the discrete measure), and I do not know if the reverse holds.
3. I think that the third implication is actually an equivalence, but I am not sure.
4. The fourth implication is proved in a paper by Janusz Pawlikowski in Fund. Math. 138 (1991). (The paper also mentions or perhaps proves the third implication.)

So the answer is: No, you cannot prove AC from Hahn-Banach. But yes, if you get rid of Banach-Tarski, you also lose Hahn-Banach.

Aleph4 23:29, 20 April 2006 (UTC)

redirect?

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)

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)

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)

infinite sub-linear function

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)

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)

fields

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)

TeXify math?

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)

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)

Eduard Helly

The story of Eduard Helly proving Hahn-Banach 20 years before Hahn and Banach seems to be supported by Mac Tutor [1]. I suppose we should add it. -- Tiphareth (talk) 11:45, 19 November 2007 (UTC)

Most general formulation

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)

Applications

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)

Domination by convex functional

There is a more general formulation of the Hahn-Banach theorem, which is hardly found in any textbooks on functional analysis and only demanded for ${\mathcal {N}}$ to be convex. It is clear that every sublinear function by

$\forall x,y\in V\forall \lambda \in [0,1]:f(\lambda x+(1-\lambda )y)\leq f(\lambda x)+f((1-\lambda )y)=\lambda f(x)+(1-\lambda )f(y)$

is convex.

Hence the most general formulation of Hahn-Banach in the real case is the following theorem from the paper A Remark Concerning Hahn-Banach's Extension Theorem and the Quasilinearization of Convex Functionals by L. Bittner (1971):

The Hahn–Banach theorem states that if $\scriptstyle {\mathcal {N}}:\;V\rightarrow [0,\infty )$ is a convex functional, and $\scriptstyle \varphi :\;U\rightarrow \mathbb {R}$ is a linear functional on a linear subspace U ⊆ V which is dominated by $\scriptstyle {\mathcal {N}}$ on U,

$\varphi (x)\leq {\mathcal {N}}(x)\qquad \forall x\in U$

then there exists a linear extension $\scriptstyle \psi :\;V\rightarrow \mathbb {R}$ of $\varphi$ to the whole space V, i.e., there exists a linear functional ψ such that

$\psi (x)=\varphi (x)\qquad \forall x\in U$

and

$\psi (x)\leq {\mathcal {N}}(x)\qquad \forall x\in V.$

But note, that the convex version, indeed need the assumption, that $\mathcal{N}$ is sublinear. Balkenbrenner (talk) 22:02, 8 November 2011 (UTC)

I think this is precisely the version of HBT given in Aliprantis-Border: Infinite dimensional analysis, p.195, Theorem 5.53. I tried searching google books for phrases like "hahn-banach theorem" "dominated by a convex" or "hahn-banach theorem" "convex function" and found this book: Foundations of optimization By Osman Güler, Graduate Texts in Mathematics 258 gives both formulations, In Theorem 6.40 on p.167 the dominating functional is sublinear, in Theorem 6.41 on p.168 the dominating functional is convex. I do not have the access to that paper now, but if my notes are correct, the version with convex function appears also in J. D. Weston: A Note on the Extension of Linear Funtionals; The American Mathematical Monthly, Vol. 67, No. 5 (May, 1960), pp. 444-445 [2]. --Kompik (talk) 17:46, 28 December 2011 (UTC)

You are right, Weston's result is:

Let g be a convex functional on a real vector space X, let X0 be a vector subspace

of X, and let f0 be a linear functional on X0 such that f0(x) < g(x) for all x in Xo. Then there is a linear functional f on X such that f(x) =f0(x) for all x in Xo, and

f(x) <g(x) for all x in X

If you wish, I can send you Weston's article. Sasha (talk) 20:39, 28 December 2011 (UTC)

Sasha there's no need for sending me the article, I'll be able to access jstor when I'm back at my workplace (in a few days). But thanks for the quote. --Kompik (talk) 06:46, 29 December 2011 (UTC)

The only assumption required is that the functional is convex. No extra assumptions on the functional (e.g. positivity, positive homogeneity, subadditivity) are not needed. See Eric Schechter, Handbook of Analysis and Its Foundations, Chapter 12, Academic Press, 1996. ISBN 0-12-622760-8. I've made the necessary edit. -- David Pal (talk) 14:43, 26 April 2018 (UTC)

False main result with sublinear functions instead of seminorms

I think the main theorem (the first stated version of the Hahn-Banach theorem) with the bound $|f|\leq p$, 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 $|F|\leq p$, 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 $f\leq p$ on $M$ and $F\leq p$ on $V$, see Theorem 3.2 in Rudin. 188.36.117.186 (talk) 17:05, 23 June 2021 (UTC)

Problematic PDE section

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 (u) 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)

"Mazur's theorem" listed at Redirects for discussion

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 Wikipedia:Redirects for discussion/Log/2024 February 13 § Mazur's theorem until a consensus is reached. Jay 💬 13:15, 13 February 2024 (UTC)