Mathematics of general relativity - Revision history

en>Dresdnhope: internal link clean up

2014-12-18T00:58:54Z

internal link clean up

← Older revision		Revision as of 02:58, 18 December 2014
Line 1:		Line 1:
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	~~== ニューバランス ==~~

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en>Bhny: /* Tensors in General Relativity */ fixed link to Preferred frame

2014-02-12T01:19:40Z

Tensors in General Relativity: fixed link to Preferred frame

← Older revision		Revision as of 03:19, 12 February 2014
Line 1:		Line 1:
	~~{{Unreferenced\|date~~=~~December 2009}}~~		== 江蘭の専門職 ==
	In [[functional analysis]] and related areas of [[mathematics]] a '''dual topology''' is a [[locally convex topology]] on a [[dual pair]], two [[vector space]]s with a [[bilinear form]] defined on them, so that one vector space becomes the [[continuous dual]] of the other space.

	~~The different dual topologies for a given dual pair are characterized by the Mackey–Arens theorem~~. ~~All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology~~.		余裕がないことはできますか？ '<br><br>玄Xiがアップ眉をひそめ、緑色の粒子を、見た。<br><br>「風水ゆう、あなたは、攻撃を受けて、この緑の粒子が気づいていないではなく、さらに場所の筋肉を動かしていませんでした。」玄Xiは疑わしい（キス小説ネットワーク [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタン店舗東京] - [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_11.php クリスチャンルブタン取扱店] メンバーの手）外を見る混乱の場所を。<br><br>ゆう黄も、このに留意した。<br>さらに悪い防衛を<br>が、その後、強い地上攻撃により、少なくとも飛ぶヒット？しかし、たとえグリーン粒子の位置はbudgedていません。残念ながら...... Xiのゆう黄とどの江蘭セクターを理解するための神秘的で強力な場所。<br>彼らの攻撃を<br>？<br><br>でも10倍悪くはその後、江蘭部門は簡単に攻撃を持ち上げることができる。<br><br>ウッドウッドウッド<br><br>江蘭の専門職 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタンメンズ]。<br><br>秦Yuは静かな栽培にまだあるが、今秦Yuは、ほぼ200年練習してきた、とさえ赤ちゃん子牛が、4天使レルムにも息の練習で、まだ秦ゆうで練習することができる [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_7.php クリスチャンルブタン銀座]。<br>魂へ<br>魂第三層 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_12.php クリスチャンルブタン値段]。<br><br>秦Yuはあります
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	~~Several topological properties depend only on the [[dual pair]] and not on the chosen dual topology and thus it is often possible to substitute a complicated dual topology by a simpler one.~~		== ニューバランス ==

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	~~Given a dual pair~~ <~~math~~>~~(X, Y, \langle , \rangle)~~</~~math~~>~~, a '''dual topology''' on~~ <~~math~~>X<~~/math~~> ~~is a [[locally convex topology]]~~ <~~math~~>~~\tau~~<~~/math~~> ~~so that~~		相关的主题文章：
	:<~~math~~>~~(X, \tau)' \simeq Y~~.</~~math>~~		<ul>
	~~Here~~ <~~math~~>~~(X, \tau)'~~<~~/math~~> ~~denotes the~~ [~~[continuous dual]~~] of <~~math~~>~~(X,\tau)~~</~~math> and~~ <~~math~~>~~(X, \tau)' \simeq Y~~<~~/math~~> ~~means that there is a [~~[~~linear isomorphism]]~~
	:~~<math>\Psi : Y \to (X, \tau)',\quad y \mapsto (x \mapsto \langle x, y\rangle)~~.<~~/math~~>		<li>[http://bbs.clzsjcw.com/forum.php?mod=viewthread&tid=190046 http://bbs.clzsjcw.com/forum.php?mod=viewthread&tid=190046]</li>
	~~(If a locally convex topology~~ <~~math~~>~~\tau~~<~~/math~~> on <~~math~~>X<~~/math~~> ~~is not a dual topology, then either~~ <~~math~~>~~\Psi~~<~~/math~~> ~~is not surjective or it is ill-defined since the linear functional~~ <~~math~~>~~x \mapsto \langle x, y\rangle<~~/~~math> is not continuous on <math>X<~~/~~math> for some <math>y<~~/~~math>~~.)
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	==~~Properties==~~
	* '''Theorem''' (by [[George Mackey\|Mackey]]): ~~Given a dual pair, the [[bounded set (topological vector space)\|bounded set]]s under any dual topology are identical~~.		<li>[http://www.s602.cn/bbs/home.php?mod=space&uid=131559 http://www.s602.cn/bbs/home.php?mod=space&uid=131559]</li>

	* Under any dual topology the same sets are [[barrelled set\|barrelled]].		</ul>

	==~~Characterization of dual topologies==~~
	~~The '''Mackey–Arens theorem''', named after [[George Mackey~~]~~] and [[Richard Friedrich Arens\|Richard Arens]], characterizes all possible dual topologies on a [[locally convex space]]s.~~

	~~The theorem shows that the [[coarser topology\|coarsest]] dual topology is the [[weak topology]], the topology of uniform convergence on all finite subsets of~~ <~~math~~>~~X'<~~/~~math>, and the [[finer topology\|finest topology]~~] ~~is the [[Mackey topology]], the topology of uniform convergence on all weakly compact subsets of <math>X'~~</~~math~~>.

	~~===Mackey–Arens theorem===~~
	~~Given a [[dual pair]]~~ <~~math~~>~~(X, X')<~~/~~math> with <math>X<~~/~~math> a locally convex~~ space ~~and <math>X'<~~/~~math> its [[continuous dual]] then <math>\tau<~~/~~math> is a dual topology on <math>X<~~/~~math> [[if and only if]] it is a [[topology of uniform convergence]] on a family of [[absolutely convex]~~] ~~and [[weak topology\|weakly compact]] subsets of~~ <~~math~~>X'</~~math~~>

	~~{{Functional Analysis}}~~

	~~{{DEFAULTSORT:Dual Topology}}~~
	~~[[Category:Topology of function spaces]]~~

en>YohanN7: Undid revision 580648989 by 173.226.31.66 (talk)

2013-11-08T15:40:14Z

Undid revision 580648989 by 173.226.31.66 (talk)

← Older revision		Revision as of 17:40, 8 November 2013
Line 1:		Line 1:
	~~They call me Emilia~~. ~~Hiring~~ is ~~her working day occupation now~~ and ~~she will~~ not ~~change~~ it ~~anytime soon. One of the things he enjoys most is ice skating but he~~ is ~~having difficulties~~ to ~~discover time for it~~. ~~South Dakota~~ is ~~where me~~ and ~~my husband reside.~~<br><br>~~Here~~ is ~~my weblog~~ :: [~~http~~://~~www~~.~~myprgenie.com~~/~~view-publication~~/~~weight-loss-tips-that-will-reduce-your-waistline weight loss food delivery~~]		{{Unreferenced\|date=December 2009}}
			In [[functional analysis]] and related areas of [[mathematics]] a '''dual topology''' is a [[locally convex topology]] on a [[dual pair]], two [[vector space]]s with a [[bilinear form]] defined on them, so that one vector space becomes the [[continuous dual]] of the other space.

			The different dual topologies for a given dual pair are characterized by the Mackey–Arens theorem. All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology.

			Several topological properties depend only on the [[dual pair]] and not on the chosen dual topology and thus it is often possible to substitute a complicated dual topology by a simpler one.

			==Definition==
			Given a dual pair <math>(X, Y, \langle , \rangle)</math>, a '''dual topology''' on <math>X</math> is a [[locally convex topology]] <math>\tau</math> so that
			:<math>(X, \tau)' \simeq Y.</math>
			Here <math>(X, \tau)'</math> denotes the [[continuous dual]] of <math>(X,\tau)</math> and <math>(X, \tau)' \simeq Y</math> means that there is a [[linear isomorphism]]
			:<math>\Psi : Y \to (X, \tau)',\quad y \mapsto (x \mapsto \langle x, y\rangle).</math>
			(If a locally convex topology <math>\tau</math> on <math>X</math> is not a dual topology, then either <math>\Psi</math> is not surjective or it is ill-defined since the linear functional <math>x \mapsto \langle x, y\rangle</math> is not continuous on <math>X</math> for some <math>y</math>.)

			==Properties==
			* '''Theorem''' (by [[George Mackey\|Mackey]]): Given a dual pair, the [[bounded set (topological vector space)\|bounded set]]s under any dual topology are identical.

			* Under any dual topology the same sets are [[barrelled set\|barrelled]].

			==Characterization of dual topologies==
			The '''Mackey–Arens theorem''', named after [[George Mackey]] and [[Richard Friedrich Arens\|Richard Arens]], characterizes all possible dual topologies on a [[locally convex space]]s.

			The theorem shows that the [[coarser topology\|coarsest]] dual topology is the [[weak topology]], the topology of uniform convergence on all finite subsets of <math>X'</math>, and the [[finer topology\|finest topology]] is the [[Mackey topology]], the topology of uniform convergence on all weakly compact subsets of <math>X'</math>.

			===Mackey–Arens theorem===
			Given a [[dual pair]] <math>(X, X')</math> with <math>X</math> a locally convex space and <math>X'</math> its [[continuous dual]] then <math>\tau</math> is a dual topology on <math>X</math> [[if and only if]] it is a [[topology of uniform convergence]] on a family of [[absolutely convex]] and [[weak topology\|weakly compact]] subsets of <math>X'</math>

			{{Functional Analysis}}

			{{DEFAULTSORT:Dual Topology}}
			[[Category:Topology of function spaces]]

87.254.71.174 at 14:15, 23 August 2012

2012-08-23T14:15:01Z

New page

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