Talk:M. Riesz extension theorem

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Sup

In the proof, why must the sup be finite?

You first choose x' such that x'-y belongs to K. This is possible by assumption. Now whenever y-x is in K, then so is x'-x, and so the sup is smaller than \phi(x'). — Preceding unsigned comment added by 78.53.96.139 (talk) 14:22, 13 April 2012 (UTC)

Theorem

The theorem as stated in this article is wrong. A counterexample can already be found in 2-space, taking K to be the upper halfplane with the negative x-axis removed. If F is the x-axis, then the positive functional φ=X can not be extended to a positive functional on the plane.

There are also counterexamples with closed convex cones, starting in dimension three.

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Thanks for pointing this out. There are probably a couple of research papers out there, which are just citing Wikipedia... —Preceding unsigned comment added by Andreas thom (talk • contribs) 18:56, 2 March 2010 (UTC)

"If F is the closed positive x-axis" -> "If F is the x-axis"

correct as stated in the article. x is not a positive functional on the whole axis, this is exactly the point of the example.

"the additional assumption that for every $y\in E$ there exist $x,x'\in F$ such that $x-y\in K$ and $y-x'\in K.$" seems redundant, and equivalent to $E=K+F$

I am not sure I understand your comment (K is not a linear space). I guess one side is sufficient, but is this a reason for an expert tag?

It is fullfilled for F=the x-axis in $\mathbb {R} ^{2}$ and $K=\{(u,v);|v|\leq u\}$, but for $y=(0,1)$, it is not true that "Every point in $E\setminus F$ is a positive linear multiple of either $x-y$ or $y-x'$ for some $x,x'\in K$"

At first sight (very quick and superficial), I was not able to locate this theorem in "M.Riesz, Sur le problème des moments". Can you please give a page number ?

Anne Bauval (talk) 09:59, 17 September 2011 (UTC)

Try the third paper, page 2.

Sasha (talk) 16:24, 17 September 2011 (UTC)

"If F is the closed positive x-axis", to me (and probably other readers), means $F=\mathbb {R} ^{+}\times \{0\}$, which is not a linear subspace

My rewriting of "the additional assumption" is :

$(\forall y\in E,\exists x\in F,x-y\in K)\Leftrightarrow E\subset F-K$ and

$(\forall y\in E,\exists x'\in F,y-x'\in K)\Leftrightarrow E\subset F+K$, and (using only that E and F are linear spaces, of course not K)

$E\subset F-K\Leftrightarrow -E\subset -F+K\Leftrightarrow E\subset F+K$

I really think that E=K+F is a simpler formulation than "for every $y\in E$ there exist $x,x'\in F$ such that $x-y\in K$ and $y-x'\in K$", but I admit it is a matter of taste.

As indicated in the comment of my edit here, the main reason for the expert tag was the flaw in the proof (see counterexample above).

Anne Bauval (talk) 20:45, 17 September 2011 (UTC)

I am sorry -- I answered in a hurry, you are right indeed. I will try to clean it up a bit. Thanks! Sasha (talk) 21:51, 17 September 2011 (UTC)