Bornological space: Difference between revisions
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In [[mathematics]], particularly in [[functional analysis]], a '''Mackey space''' is a [[locally convex topological vector space]] ''X'' such that the [[topology]] of ''X'' coincides with the [[Mackey topology]] τ(''X'',''X′''), the [[finer topology|finest topology]] which still preserves the [[continuous dual]]. | |||
==Examples== | |||
Examples of Mackey spaces include: | |||
* All [[bornological space]]s. | |||
* All Hausdorff locally convex [[barrelled space|quasi-barrelled]] (and hence all Hausdorff locally convex [[barrelled space]]s and all Hausdorff locally convex reflexive spaces). | |||
* All Hausdorff locally convex [[metrizable space]]s. | |||
==Properties== | |||
* A locally convex space <math>X</math> with continuous dual <math>X'</math> is a Mackey space if and only if each convex and <math>\sigma(X', X)</math>-relatively compact subset of <math>X'</math> is equicontinuous. | |||
* The [[Completion (metric space)|completion]] of a Mackey space is again a Mackey space. | |||
* A separated quotient of a Mackey space is again a Mackey space. | |||
* A Mackey space need not be separable, complete, quasi-barrelled, nor <math>\sigma</math>-quasi-barrelled. | |||
== References == | |||
* {{cite book |last=Robertson |first=A.P. |coauthors= W.J. Robertson |title= Topological vector spaces |series=Cambridge Tracts in Mathematics |volume=53 |year=1964 |publisher= [[Cambridge University Press]] | page=81 }} | |||
* {{cite book | author=H.H. Schaefer | title=Topological Vector Spaces | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics|GTM]] | volume=3 | date=1970 | isbn=0-387-05380-8 | pages=132–133 }} | |||
* {{cite book | author=S.M. Khaleelulla | title=Counterexamples in Topological Vector Spaces | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics|GTM]] | volume=936 | date=1982 | isbn=978-3-540-11565-6 | pages=31, 41, 55-58 }} | |||
{{Functional Analysis}} | |||
[[Category:Topological vector spaces]] | |||
{{mathanalysis-stub}} | |||
Revision as of 21:11, 23 January 2014
In mathematics, particularly in functional analysis, a Mackey space is a locally convex topological vector space X such that the topology of X coincides with the Mackey topology τ(X,X′), the finest topology which still preserves the continuous dual.
Examples
Examples of Mackey spaces include:
- All bornological spaces.
- All Hausdorff locally convex quasi-barrelled (and hence all Hausdorff locally convex barrelled spaces and all Hausdorff locally convex reflexive spaces).
- All Hausdorff locally convex metrizable spaces.
Properties
- A locally convex space with continuous dual is a Mackey space if and only if each convex and -relatively compact subset of is equicontinuous.
- The completion of a Mackey space is again a Mackey space.
- A separated quotient of a Mackey space is again a Mackey space.
- A Mackey space need not be separable, complete, quasi-barrelled, nor -quasi-barrelled.
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534