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| In [[functional analysis]], a branch of [[mathematics]], the '''strong operator topology''', often abbreviated SOT, is the weakest [[locally convex]] [[topology]] on the set of [[bounded operator]]s on a [[Hilbert space]] (or, more generally, on a [[Banach space]]) such that the evaluation map sending an operator ''T'' to the real number <math>\|Tx\|</math> is [[continuous function (topology)|continuous]] for each vector ''x'' in the Hilbert space. | | Human being who wrote the commentary is called Roberto Ledbetter and his wife go like it at every bit. In his professional life he is also a people [http://Answers.Yahoo.com/search/search_result?p=manager&submit-go=Search+Y!+Answers manager]. He's always loved living to Guam and he has everything that he needs there. The favorite hobby for him also his kids is growing plants but he's been [https://www.Google.com/search?hl=en&gl=us&tbm=nws&q=ingesting ingesting] on new things lately. He's been working on any website for some era now. Check it out here: http://circuspartypanama.com<br><br>my homepage; [http://circuspartypanama.com clash of clans Cheats deutsch] |
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| The SOT is [[finer topology|stronger]] than the [[weak operator topology]] and weaker than the [[operator norm|norm topology]].
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| The SOT lacks some of the nicer properties that the [[weak operator topology]] has, but being stronger, things are sometimes easier to prove in this topology. It is more natural too, since it is simply the topology of pointwise convergence for an operator. | |
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| The SOT topology also provides the framework for the [[measurable functional calculus]], just as the norm topology does for the [[continuous functional calculus]].
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| The [[linear functional]]s on the set of bounded operators on a Hilbert space that are continuous in the SOT are precisely those continuous in the [[weak operator topology|WOT]]. Because of this, the closure of a [[convex set]] of operators in the WOT is the same as the closure of that set in the SOT.
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| It should also be noted that the above language translates into convergence properties of Hilbert space operators. One especially observes that for a complex Hilbert space, by way of the polarization identity, one easily verifies that Strong Operator convergence implies Weak Operator convergence.
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| ==See also== | |
| *[[Strongly continuous semigroup]]
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| *[[Topologies on the set of operators on a Hilbert space]]
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| ==References==
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| *{{cite book |last=Rudin |first=Walter |title=Functional Analysis |date=January 1991 |publisher=McGraw-Hill Science/Engineering/Math |isbn=0-07-054236-8}}
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| *{{cite book |last=Pedersen |first=Gert |title=Analysis Now |year=1989 |publisher=Springer |isbn=0-387-96788-5}}
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| {{Functional Analysis}}
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| [[Category:Topology of function spaces]]
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