Hilbert's seventh problem - Revision history

86.185.237.242: /* See also */ Gelfond–Schneider constant

2015-01-10T00:35:25Z

35.13.68.198 at 16:17, 19 February 2014

2014-02-19T16:17:27Z

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	~~In [[mathematics]], a '''rigged Hilbert space''' ('''Gelfand triple''', '''nested Hilbert space''', '''equipped Hilbert space''')~~ is ~~a construction designed~~ to ~~link~~ the ~~[[distribution (mathematics)\|distribution]] and [[square-integrable]] aspects of [[functional analysis]]~~. ~~Such spaces were introduced~~ to ~~study~~ [~~[spectral theory]~~] in ~~the broad sense~~.~~{{vague\|~~It ~~does something that isn~~'~~t everything\|date=January 2012}} They can bring together the '[[bound state]]' ([[eigenvector]])~~ and '~~[[Decomposition of spectrum (functional analysis)\|continuous spectrum]]', in one place~~.		Eusebio is the name women and men use to call for me and I think the problem sounds quite good when you say it. I worn to be unemployed but also now I am a great [https://www.vocabulary.com/dictionary/cashier cashier]. My house is from now on in South Carolina in addition I don't plan onto changing it. It's not a common concern but what I comparable to doing is bottle top collecting and now I do have time to have a look at on new things. I'm not high-quality at webdesign but you may be want to check brand new website: http://circuspartypanama.com<br><br>

	~~==Motivation==~~		Look at my page ... clash of clans hack android, [http://circuspartypanama.com http://circuspartypanama.com],

	~~A function such as the canonical homomorphism of the real line into the complex plane~~

	~~:<math> x \mapsto e^{ix} , </math>~~

	~~is an [[eigenvector]] of the [[differential operator]]~~

	~~:<math>-i\frac{d}{dx}</math>~~

	on the [[real line]] '''R''', but isn't [[square-integrable]] for the usual [[Borel measure]] on '''R'''. To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the [[Hilbert space]] theory. ~~This was supplied by the apparatus of [[Schwartz distribution]]s, and a ''generalized eigenfunction'' theory was developed in the years after 1950~~.

	~~==Functional analysis approach==~~

	~~The concept of rigged Hilbert space places this idea in an abstract functional-analytic framework~~. ~~Formally, a rigged Hilbert space consists~~ of ~~a [[Hilbert space]] ''H''~~, ~~together with a subspace Φ which carries a~~ [~~[finer topology]], that is one for which the natural inclusion~~

	~~:<math> \Phi \subseteq H </math>~~

	is continuous. It is [[Without loss of generality\|no loss]] to assume that Φ is [[Dense_set\|dense]] in ''H'' for the Hilbert norm. We consider the inclusion of [[dual space]]s ''H''<sup></sup> in Φ<sup></sup>. The latter, dual to Φ in its 'test function' topology, is realised as a space of distributions or generalised functions of some sort, and the [[linear functional]]s on the subspace Φ of type

	:~~<math>\phi\mapsto\langle v,\phi\rangle</math>~~

	~~for ''v'' in ''H'' are faithfully represented as distributions (because we assume Φ dense).~~

	~~Now by applying the [[Riesz representation theorem]] we can identify ''H''<sup>*<~~/~~sup> with ''H''. Therefore the definition of ''rigged Hilbert space'' is in terms of a sandwich:~~

	~~:<math>\Phi \subseteq H \subseteq \Phi^*. <~~/~~math>~~

	The most significant examples are those for which Φ is a [[nuclear space]]; this comment is an abstract expression of the idea that Φ consists of test functions and Φ* of the corresponding [[distribution (mathematics)\|distributions]].

	~~==Formal definition (Gelfand triple)==~~

	A '''rigged Hilbert space''' is a pair (''H'',Φ) with ''H'' a Hilbert space, Φ a dense subspace, such that Φ is given a [[topological vector space]] structure for which the [[inclusion map]] ''i'' is continuous.

	~~Identifying ''H'' with its dual space ''H<sup>*</sup>'', the adjoint to ''i'' is the map~~

	~~:<math>i^:H=H^\to\Phi^*.</math>~~

	~~The duality pairing between Φ and Φ<sup>*</sup> has to be compatible with the inner product on ''H'', in the sense that:~~

	~~:<math>\langle u, v\rangle_{\Phi\times\Phi^*} = (u, v)_H</math>~~

	~~whenever <math>u\in\Phi\subset H</math> and <math>v \in H=H^* \subset \Phi^*</math>.~~

	~~The specific triple <math> (\Phi,\,\,H,\,\,\Phi^*)</math> is often named the "Gelfand triple" (after the mathematician [[Israel Gelfand]]).~~

	Note that even though Φ is isomorphic to Φ<sup></sup> if Φ is a Hilbert space in its own right, this isomorphism is ''not'' the same as the composition of the inclusion ''i'' with its adjoint ''i''

	~~:<math>i^* i:\Phi\subset H=H^\to\Phi^.</math>~~

	~~==References==~~

	* J.-P. Antoine, ''Quantum Mechanics Beyond Hilbert Space'' (1996), appearing in ''Irreversibility and Causality, Semigroups and Rigged Hilbert Spaces'', Arno Bohm, Heinz-Dietrich Doebner, Piotr Kielanowski, eds., Springer-Verlag, ISBN 3-540-64305-2. ''(Provides a survey overview.)''
	* [[Jean Dieudonné]], ''Éléments d'analyse'' VII (1978). ''(See paragraphs 23.8 and 23.32)''
	* [[Israel Gelfand\|I. M. Gelfand]] and N. J. Vilenkin. Generalized Functions, vol. 4: Some Applications of Harmonic Analysis. Rigged Hilbert Spaces. Academic Press, New York, 1964.
	* R. de la Madrid, "The role of the rigged Hilbert space in Quantum Mechanics," Eur. J. Phys. 26, 287 (2005); [http://~~arxiv~~.~~org/abs/quant-ph/0502053 quant-ph/0502053~~].
	* K. Maurin, ''Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups'', Polish Scientific Publishers, Warsaw, ~~1968.~~
	*{{eom\|id=Rigged_Hilbert_space\|first=R.A.\|last= Minlos}}

	~~[[Category:Hilbert space]]~~
	~~[[Category:Spectral theory]]~~
	~~[[Category:Generalized functions]]~~

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2013-02-26T07:40:40Z

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← Older revision		Revision as of 09:40, 26 February 2013
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			In [[mathematics]], a '''rigged Hilbert space''' ('''Gelfand triple''', '''nested Hilbert space''', '''equipped Hilbert space''') is a construction designed to link the [[distribution (mathematics)\|distribution]] and [[square-integrable]] aspects of [[functional analysis]]. Such spaces were introduced to study [[spectral theory]] in the broad sense.{{vague\|It does something that isn't everything\|date=January 2012}} They can bring together the '[[bound state]]' ([[eigenvector]]) and '[[Decomposition of spectrum (functional analysis)\|continuous spectrum]]', in one place.

			==Motivation==

	~~On top~~ of the ~~Clash of Clans hack tool; there is also hack tools to suit other games. We can check out~~ the ~~ones hacks and obtain such which they need. It~~ is ~~sure that they will have lost at fun once they own~~ the ~~hack tool available.~~<br><br>~~Beginning nearly enough pebbles to get another local building company. Don~~''t ~~waste one of~~ the ~~gems here in any way~~ on ~~rush-building anything,~~ as ~~if it can save you~~ the ~~group you are going that can eventually obtain enough free extra gems to produce that~~ [~~http://Www~~.~~Sharkbayte.com/keyword/extra+builder extra builder~~] ~~without any cost. Particularly~~, the ~~customer can get free jewelry for clearing obstructions like rocks and trees,~~ after ~~you clear them on the market they come back and you may re~~-~~clear these kind of people to get more gems~~.~~<br><br>Okazaki~~, ~~japan tartan draws incentive through your country~~'~~s adoration for cherry blossom and should include pink~~, ~~white~~, ~~green and brown lightly colours. clash of clans cheats. If you are you looking~~ for ~~more regarding [http~~:/~~/prometeu~~.~~net clash~~ of ~~clans hack download~~] ~~take a look at the web page. Style~~ is ~~called Sakura, okazaki, japan~~ for ~~cherry blossom~~.<br><br>~~On line games acquire more that can offer your son or daughter than only a possibility~~ to ~~capture points. Try deciding on adventure titles that instruct your young person some thing. In view that an example~~, ~~sports situations video games will assist your youngster learn~~ some ~~of the guidelines for game titles~~, and ~~exactly how web based games are played finally out. Check out some testimonials in discover game titles who seem to supply a learning experience instead~~ of ~~just mindless, repeated motion.~~<br><br>~~Kill time~~ for ~~game of~~ the [~~http:~~/~~/photo~~.~~net/gallery/tag-search/search?query_string=actual+season actual season] editions~~ of ~~significant titles~~. ~~These~~ most of the ~~time come out per time~~ of ~~year or higher after all~~ the ~~initial headline~~, ~~but include things like~~ a ~~lot of specific down-loadable and extra matter which was released when steps once the for~~ a ~~start headline. These ball game titles supply~~ a ~~wonderful deal more bang~~ for ~~an buck~~.<br><br>~~In order~~ to ~~some money on your games, think about following into a assistance that you can rent payments games from~~. The ~~price of these lease commitments for the year happens~~ to be ~~normally under~~ the ~~amount of two video programs. You can preserve~~ the ~~video games titles until you do better than them~~ and ~~simply transmit out them back after more and purchase another type~~.<br><br>~~Now there are~~ is a "~~start~~" ~~control~~ to ~~click on inside of~~ the ~~wake~~ of ~~entering~~ the ~~wanted traits~~. ~~When you start from Clash of Clans crack hack cheats tool~~, ~~hold around for~~ a ~~48 % of moment~~, ~~slammed refresh~~ and ~~you likely will have the means they needed~~. ~~There~~ '~~s nothing at all improper in working with thjis hack~~ and ~~cheats method~~. ~~Make utilization~~ of ~~all~~ the ~~Means that an individual have~~, and ~~exploit your 2013 Clash~~ of Clans hack obtain! For what reason fork out for cashflow or gems when yourself can get the planned things with this building! Sprint and access your proprietary Clash related to Clans hack software lately. ~~The required property are only a a number of of clicks absent~~.		A function such as the canonical homomorphism of the real line into the complex plane

			:<math> x \mapsto e^{ix} , </math>

			is an [[eigenvector]] of the [[differential operator]]

			:<math>-i\frac{d}{dx}</math>

			on the [[real line]] '''R''', but isn't [[square-integrable]] for the usual [[Borel measure]] on '''R'''. To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the [[Hilbert space]] theory. This was supplied by the apparatus of [[Schwartz distribution]]s, and a ''generalized eigenfunction'' theory was developed in the years after 1950.

			==Functional analysis approach==

			The concept of rigged Hilbert space places this idea in an abstract functional-analytic framework. Formally, a rigged Hilbert space consists of a [[Hilbert space]] ''H'', together with a subspace Φ which carries a [[finer topology]], that is one for which the natural inclusion

			:<math> \Phi \subseteq H </math>

			is continuous. It is [[Without loss of generality\|no loss]] to assume that Φ is [[Dense_set\|dense]] in ''H'' for the Hilbert norm. We consider the inclusion of [[dual space]]s ''H''<sup></sup> in Φ<sup></sup>. The latter, dual to Φ in its 'test function' topology, is realised as a space of distributions or generalised functions of some sort, and the [[linear functional]]s on the subspace Φ of type

			:<math>\phi\mapsto\langle v,\phi\rangle</math>

			for ''v'' in ''H'' are faithfully represented as distributions (because we assume Φ dense).

			Now by applying the [[Riesz representation theorem]] we can identify ''H''<sup>*</sup> with ''H''. Therefore the definition of ''rigged Hilbert space'' is in terms of a sandwich:

			:<math>\Phi \subseteq H \subseteq \Phi^*. </math>

			The most significant examples are those for which Φ is a [[nuclear space]]; this comment is an abstract expression of the idea that Φ consists of test functions and Φ* of the corresponding [[distribution (mathematics)\|distributions]].

			==Formal definition (Gelfand triple)==

			A '''rigged Hilbert space''' is a pair (''H'',Φ) with ''H'' a Hilbert space, Φ a dense subspace, such that Φ is given a [[topological vector space]] structure for which the [[inclusion map]] ''i'' is continuous.

			Identifying ''H'' with its dual space ''H<sup>*</sup>'', the adjoint to ''i'' is the map

			:<math>i^:H=H^\to\Phi^*.</math>

			The duality pairing between Φ and Φ<sup>*</sup> has to be compatible with the inner product on ''H'', in the sense that:

			:<math>\langle u, v\rangle_{\Phi\times\Phi^*} = (u, v)_H</math>

			whenever <math>u\in\Phi\subset H</math> and <math>v \in H=H^* \subset \Phi^*</math>.

			The specific triple <math> (\Phi,\,\,H,\,\,\Phi^*)</math> is often named the "Gelfand triple" (after the mathematician [[Israel Gelfand]]).

			Note that even though Φ is isomorphic to Φ<sup></sup> if Φ is a Hilbert space in its own right, this isomorphism is ''not'' the same as the composition of the inclusion ''i'' with its adjoint ''i''

			:<math>i^* i:\Phi\subset H=H^\to\Phi^.</math>

			==References==

			* J.-P. Antoine, ''Quantum Mechanics Beyond Hilbert Space'' (1996), appearing in ''Irreversibility and Causality, Semigroups and Rigged Hilbert Spaces'', Arno Bohm, Heinz-Dietrich Doebner, Piotr Kielanowski, eds., Springer-Verlag, ISBN 3-540-64305-2. ''(Provides a survey overview.)''
			* [[Jean Dieudonné]], ''Éléments d'analyse'' VII (1978). ''(See paragraphs 23.8 and 23.32)''
			* [[Israel Gelfand\|I. M. Gelfand]] and N. J. Vilenkin. Generalized Functions, vol. 4: Some Applications of Harmonic Analysis. Rigged Hilbert Spaces. Academic Press, New York, 1964.
			* R. de la Madrid, "The role of the rigged Hilbert space in Quantum Mechanics," Eur. J. Phys. 26, 287 (2005); [http://arxiv.org/abs/quant-ph/0502053 quant-ph/0502053].
			* K. Maurin, ''Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups'', Polish Scientific Publishers, Warsaw, 1968.
			*{{eom\|id=Rigged_Hilbert_space\|first=R.A.\|last= Minlos}}

			[[Category:Hilbert space]]
			[[Category:Spectral theory]]
			[[Category:Generalized functions]]

en>TakuyaMurata: lk

2012-07-22T21:17:28Z

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← Older revision		Revision as of 02:35, 10 January 2015
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