en>Yobot: WP:CHECKWIKI error fixes using AWB (9615)

2013-11-09T00:16:03Z

WP:CHECKWIKI error fixes using AWB (9615)

← Older revision		Revision as of 02:16, 9 November 2013
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	~~Friends call her Claude Gulledge~~. I'~~ve always loved living~~ in ~~Idaho~~. ~~Bottle tops gathering~~ is the ~~only hobby his wife doesn~~'~~t approve~~ of. ~~Managing people~~ is ~~how I make money~~ and it's ~~some thing I really enjoy~~.<br><br>~~Take~~ a ~~look at my webpage~~; ~~[http~~://Wp.~~shizzlaz~~.eu/~~index~~.~~php?mod~~=~~users~~&~~action~~=~~view&id~~=~~23474~~ http://Wp.~~shizzlaz~~.eu]		The '''M. Riesz extension theorem''' is a [[theorem]] in [[mathematics]], proved by [[Marcel Riesz]] <ref>{{harvtxt\|Riesz\|1923}}</ref> during his study of the [[Moment problem\|problem of moments]].<ref>{{harvtxt\|Akhiezer\|1965}}</ref>

			==Formulation==
			Let ''E'' be a [[real number\|real]] [[vector space]], ''F'' ⊂ ''E'' a [[vector subspace]], and let ''K'' ⊂ ''E'' be a [[convex cone]].

			A [[linear functional]] ''φ'': ''F'' → '''R''' is called ''K''-''positive'', if

			:<math>\phi(x) \geq 0 \quad \text{for} \quad x \in F \cap K.</math>

			A linear functional ''ψ'': ''E'' → '''R''' is called a ''K''-positive ''extension'' of ''φ'' if

			:<math>\psi\|_F = \phi \quad \text{and} \quad \psi(x) \geq 0\quad \text{for} \quad x \in K.</math>

			In general, a ''K''-positive linear functional on ''F'' can not be extended to a <math>K</math>-positive linear functional on ''E''. Already in two dimensions one obtains a counterexample taking ''K'' to be the upper halfplane with the open negative ''x''-axis removed. If ''F'' is the real axis, then the positive functional ''φ''(''x'', 0) = ''x'' can not be extended to a positive functional on the plane.

			However, the extension exists under the additional assumption that for every ''y'' ∈ ''E'' there exists ''x''∈''F'' such that ''y'' − ''x'' ∈''K''; in other words, if ''E'' = ''K'' + ''F''.

			==Proof==
			By [[transfinite induction]] it is sufficient to consider the case dim ''E''/''F'' = 1.

			Choose ''y'' ∈ ''E''\''F''. Set

			:<math>
			\psi\|_F = \phi, \quad
			\psi(y) = \sup \left\{ \phi(x) \, \mid \, x \in F, \, y - x \in K \right\},
			</math>

			and extend ''ψ'' to ''E'' by linearity. Let us show that ''ψ'' is ''K''-positive.

			Every point ''z'' in ''K'' is a positive linear multiple of either ''x'' + ''y'' or ''x'' − ''y'' for some ''x'' ∈ ''F''. In the first case, ''z'' = ''a''(''y'' + ''x''), therefore ''y''− ''(''−''x)'' = ''z''/''a''  is in  ''K''  with  −''x''  in  ''F'' . Hence

			:<math> \psi(y) \geq \psi(-x) = - \psi(x), </math>

			therefore ''ψ''(''z'') ≥ 0. In the second case, ''z'' = ''a''(''x'' − ''y''), therefore ''y'' = ''x'' − ''z''/''a''. Let ''x''<sub>1</sub> ∈ ''F'' be such that ''z''<sub>1</sub> = ''y'' − ''x''<sub>1</sub> ∈ ''K'' and ''ψ''(''x''<sub>1</sub>) ≥ ψ(''y'') − ''ε''. Then

			:<math> \psi(x) - \psi(x_1) = \psi(x-x_1) = \psi(z_1 + z/a) = \phi(z_1 + z/a) \geq 0~, </math>

			therefore ''ψ''(''z'') ≥ −''a'' ''ε''. Since this is true for arbitrary ''ε'' > 0, we obtain ''ψ''(''z'') ≥ 0.

			==Corollary: Krein's extension theorem==

			Let ''E'' be a [[real number\|real]] [[linear space]], and let ''K'' ⊂ ''E'' be a [[convex cone]]. Let ''x'' ∈ ''E''\(−''K'') be such that '''R''' ''x'' + ''K'' = ''E''. Then there exists a ''K''-positive linear functional ''φ'': ''E'' → '''R''' such that ''φ''(''x'') > 0.

			==Connection to the Hahn–Banach theorem==
			{{main\|Hahn–Banach theorem}}

			The Hahn–Banach theorem can be deduced from the M. Riesz extension theorem.

			Let ''V'' be a linear space, and let ''N'' be a sublinear function on ''V''. Let ''φ'' be a functional on a subspace ''U'' ⊂ ''V'' that is dominated by ''N'':

			:<math> \phi(x) \leq N(x), \quad x \in U.</math>

			The Hahn–Banach theorem asserts that ''φ'' can be extended to a linear functional on ''V'' that is dominated by ''N''.

			To derive this from the M. Riesz extension theorem, define a convex cone ''K'' ⊂ '''R'''×''V'' by

			:<math> K = \left\{ (a, x) \, \mid \, N(x) \leq a \right\}.</math>

			Define a functional ''φ''<sub>1</sub> on '''R'''×''U'' by

			:<math> \phi_1(a, x) = a - \phi(x).</math>

			One can see that ''φ''<sub>1</sub> is ''K''-positive, and that ''K'' + ('''R''' × ''U'') = '''R''' × ''V''. Therefore ''φ''<sub>1</sub> can be extended to a ''K''-positive functional ''ψ''<sub>1</sub> on '''R'''×''V''. Then

			:<math> \psi(x) = - \psi_1(0, x) </math>

			is the desired extension of ''φ''. Indeed, if ''ψ''(''x'') > ''N''(''x''), we have: (''N''(''x''), ''x'') ∈ ''K'', whereas

			:<math> \psi_1(N(x), x) = N(x) - \psi(x) < 0, </math>

			leading to a contradiction.

			==Notes==
			{{Reflist}}

			==References==
			* {{Citation\|last=Castillo\|first=Reńe E.\|title=A note on Krein’s theorem\|url=http://www.scm.org.co/aplicaciones/revista/Articulos/767.pdf\|journal=Lecturas Matematicas\|volume=26\|year=2005}}
			* {{Citation\|jfm=49.0195.01\|last=Riesz\|first=M.\|title=Sur le problème des moments. III.\|language=French\|journal=Ark. F. Mat. Astr. O. Fys.\|volume=17\|issue=16\|year=1923}}
			* {{Citation\|mr=0184042\|first=N.I.\|last=Akhiezer\|title=The classical moment problem and some related questions in analysis\|publisher=Hafner Publishing Co.\|location=New York\|year=1965}}

			{{Functional Analysis}}

			[[Category:Theorems in convex geometry]]
			[[Category:Theorems in functional analysis]]

202.78.240.7: /* Ballot inspection */ as...than -> as...as

2012-04-27T06:52:43Z

Ballot inspection: as...than -> as...as

New page

Friends call her Claude Gulledge. I've always loved living in Idaho. Bottle tops gathering is the only hobby his wife doesn't approve of. Managing people is how I make money and it's some thing I really enjoy.<br><br>Take a look at my webpage; [http://Wp.shizzlaz.eu/index.php?mod=users&action=view&id=23474 http://Wp.shizzlaz.eu]

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en>Yobot: WP:CHECKWIKI error fixes using AWB (9615)

202.78.240.7: /* Ballot inspection */ as...than -> as...as