MINOS - Revision history

en>Tom.Reding: WP:GenFixes (page/s, endash, , unicodify, and/or concising wikilinks, etc.), et al., ed., ref cleanup using AWB

2015-01-12T19:22:09Z

WP:GenFixes (page/s, endash, &nbsp, unicodify, and/or concising wikilinks, etc.), et al., ed., ref cleanup using AWB

← Older revision		Revision as of 21:22, 12 January 2015
Line 1:		Line 1:
	~~In [[functional analysis]], the~~ '''ultrastrong topology''', or '''σ-strong topology''', or '''strongest topology''' on the set ''B(H)'' of [[bounded operator]]s on a [[Hilbert space]] is the topology defined by the family of seminorms		Jayson Berryhill is how I'm known as and my spouse doesn't like it at all. The favorite hobby for him and his children is to perform lacross and he would by no means give it up. Office supervising is what she does for a living. North Carolina is the location he loves most but now he is considering other options.<br><br>Look into my weblog :: [http://brazil.amor-amore.com/irboothe online psychic reading]
	~~:<math>~~
	~~p_\omega(x) = \omega(x^{*} x)^{1/2}~~
	~~</math>~~
	~~for positive elements <math>\omega</math> of the [[predual]] <math>L_{*}(H)</math> that consists of trace class operators~~.
	~~<ref name="TakesakiI">{{Cite book~~
	~~\|last=Takesaki~~
	~~\|first=Masamichi~~
	~~\|title=Theory of operator algebras. I.~~
	~~\|publisher=Springer-Verlag\|location=[[Berlin]]~~
	~~\|year=2002~~
	~~\|isbn=3-540-42248-X~~
	~~}}</ref>{{rp\|68}}~~

	~~It was introduced by [[von Neumann]] in 1936.~~
	~~<ref>{{Citation~~
	~~\| last=von Neumann~~
	~~\| first=John~~
	~~\| title=On a Certain Topology~~ for ~~Rings of Operators~~
	~~\| journal=Annals of Mathematics. Second Series~~
	~~\| volume=37~~
	~~\| issue=1~~
	~~\| year=1936~~
	~~\| pages=111–115~~
	~~\| url=http://links.jstor.org/sici?sici=0003-486X%28193601%292%3A37%3A1%3C111%3AOACTFR%3E2.0.CO%3B2-S~~
	~~}}</ref>~~

	~~==Relation with the strong (operator) topology==~~

	~~The ultrastrong topology~~ is ~~similar~~ to ~~the strong (operator) topology.~~
	~~For example, on any norm-bounded set the strong operator~~ and ~~ultrastrong topologies~~
	~~are the same~~. ~~The ultrastrong topology~~ is ~~stronger than the strong operator topology~~.

	~~One problem with the strong operator topology~~ is ~~that~~ the ~~dual of ''B(H)'' with the strong operator topology is~~
	~~"too small". The ultrastrong topology fixes this problem: the dual is the full predual~~
	~~''B<sub>*</sub>(H)'' of all trace class operators. In general the ultrastrong topology is better than the strong operator topology,~~ but is ~~more complicated to define so people usually use the strong operator topology if they can get away with it.~~

	~~The ultrastrong topology can be obtained from the strong operator topology as follows~~.
	~~If ''H''~~<~~sub~~>1<~~/sub~~> ~~is a separable infinite dimensional Hilbert space~~
	~~then ''B(H)'' can be embedded in ''B''(''H''&otimes;''H''<sub>1</sub>) by tensoring with the identity map on ''H''<sub>1</sub>. Then the restriction of the strong operator topology on~~
	~~''B''(''H''&otimes;''H''<sub>1</sub>) is the ultrastrong topology of ''B(H)''.~~
	~~Equivalently, it is given by the family of seminorms~~
	:~~<math>~~
	~~x \mapsto \left( \sum_{n=1}^\infty \|\|x\xi_n\|\|^2 \right)^{1~~/~~2},~~
	</~~math>~~
	~~where <math>\sum_{n=1}^\infty \|\|\xi_n\|\|^2 < \infty</math>~~.~~<ref name=TakesakiI/>{{rp\|68}}~~

	~~The adjoint map is not continuous in the ultrastrong topology~~. ~~There is another topology called the ultrastrong<sup>*<~~/~~sup> topology, which is the weakest topology stronger than the ultrastrong topology such that the adjoint map is continuous.<ref name=TakesakiI/>{{rp\|68}}~~

	~~== See also ==~~
	*[[Topologies on the set of operators on a Hilbert space]]
	*[[ultraweak topology]]
	*[[strong operator topology]]

	~~==References==~~
	~~{{Reflist}}~~

	~~{{Functional Analysis}}~~

	~~[[Category:Topology of function spaces]]~~
	~~[[Category:Von Neumann algebras]~~]

en>BattyBot: /* Physics goals and results */fixed CS1 errors: dates & General fixes using AWB (9832)

2014-01-03T21:19:56Z

Physics goals and results: fixed CS1 errors: dates & General fixes using AWB (9832)

← Older revision		Revision as of 23:19, 3 January 2014
Line 1:		Line 1:
	~~The writer~~'s name is ~~Christy Brookins~~. ~~To perform lacross~~ is the ~~factor I adore most~~ of all. ~~For a while I've been in Alaska~~ but ~~I will have~~ to ~~transfer in a yr or two~~. ~~Invoicing~~ is ~~my occupation~~.<br><br>~~Also visit my blog post: authentic psychic readings [[http~~://~~kjhkkb~~.~~net~~/xe/~~notice~~/~~374835 hop over to this website~~]]		In [[functional analysis]], the '''ultrastrong topology''', or '''σ-strong topology''', or '''strongest topology''' on the set ''B(H)'' of [[bounded operator]]s on a [[Hilbert space]] is the topology defined by the family of seminorms
			:<math>
			p_\omega(x) = \omega(x^{*} x)^{1/2}
			</math>
			for positive elements <math>\omega</math> of the [[predual]] <math>L_{*}(H)</math> that consists of trace class operators.
			<ref name="TakesakiI">{{Cite book
			\|last=Takesaki
			\|first=Masamichi
			\|title=Theory of operator algebras. I.
			\|publisher=Springer-Verlag\|location=[[Berlin]]
			\|year=2002
			\|isbn=3-540-42248-X
			}}</ref>{{rp\|68}}

			It was introduced by [[von Neumann]] in 1936.
			<ref>{{Citation
			\| last=von Neumann
			\| first=John
			\| title=On a Certain Topology for Rings of Operators
			\| journal=Annals of Mathematics. Second Series
			\| volume=37
			\| issue=1
			\| year=1936
			\| pages=111–115
			\| url=http://links.jstor.org/sici?sici=0003-486X%28193601%292%3A37%3A1%3C111%3AOACTFR%3E2.0.CO%3B2-S
			}}</ref>

			==Relation with the strong (operator) topology==

			The ultrastrong topology is similar to the strong (operator) topology.
			For example, on any norm-bounded set the strong operator and ultrastrong topologies
			are the same. The ultrastrong topology is stronger than the strong operator topology.

			One problem with the strong operator topology is that the dual of ''B(H)'' with the strong operator topology is
			"too small". The ultrastrong topology fixes this problem: the dual is the full predual
			''B<sub>*</sub>(H)'' of all trace class operators. In general the ultrastrong topology is better than the strong operator topology, but is more complicated to define so people usually use the strong operator topology if they can get away with it.

			The ultrastrong topology can be obtained from the strong operator topology as follows.
			If ''H''<sub>1</sub> is a separable infinite dimensional Hilbert space
			then ''B(H)'' can be embedded in ''B''(''H''&otimes;''H''<sub>1</sub>) by tensoring with the identity map on ''H''<sub>1</sub>. Then the restriction of the strong operator topology on
			''B''(''H''&otimes;''H''<sub>1</sub>) is the ultrastrong topology of ''B(H)''.
			Equivalently, it is given by the family of seminorms
			:<math>
			x \mapsto \left( \sum_{n=1}^\infty \|\|x\xi_n\|\|^2 \right)^{1/2},
			</math>
			where <math>\sum_{n=1}^\infty \|\|\xi_n\|\|^2 < \infty</math>.<ref name=TakesakiI/>{{rp\|68}}

			The adjoint map is not continuous in the ultrastrong topology. There is another topology called the ultrastrong<sup>*</sup> topology, which is the weakest topology stronger than the ultrastrong topology such that the adjoint map is continuous.<ref name=TakesakiI/>{{rp\|68}}

			== See also ==
			*[[Topologies on the set of operators on a Hilbert space]]
			*[[ultraweak topology]]
			*[[strong operator topology]]

			==References==
			{{Reflist}}

			{{Functional Analysis}}

			[[Category:Topology of function spaces]]
			[[Category:Von Neumann algebras]]

en>Bibcode Bot: Adding 2 arxiv eprint(s), 0 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

2012-08-20T04:09:04Z

Adding 2 arxiv eprint(s), 0 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

New page

The writer's name is Christy Brookins. To perform lacross is the factor I adore most of all. For a while I've been in Alaska but I will have to transfer in a yr or two. Invoicing is my occupation.<br><br>Also visit my blog post: authentic psychic readings [[http://kjhkkb.net/xe/notice/374835 hop over to this website]]