Strong dual space

Summary

In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) is the continuous dual space of equipped with the strong (dual) topology or the topology of uniform convergence on bounded subsets of where this topology is denoted by or The coarsest polar topology is called weak topology. The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise. To emphasize that the continuous dual space, has the strong dual topology, or may be written.

Strong dual topology

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Throughout, all vector spaces will be assumed to be over the field   of either the real numbers   or complex numbers  

Definition from a dual system

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Let   be a dual pair of vector spaces over the field   of real numbers   or complex numbers   For any   and any   define

 

Neither   nor   has a topology so say a subset   is said to be bounded by a subset   if   for all   So a subset   is called bounded if and only if

 
This is equivalent to the usual notion of bounded subsets when   is given the weak topology induced by   which is a Hausdorff locally convex topology.

Let   denote the family of all subsets   bounded by elements of  ; that is,   is the set of all subsets   such that for every  

 
Then the strong topology   on   also denoted by   or simply   or   if the pairing   is understood, is defined as the locally convex topology on   generated by the seminorms of the form
 

The definition of the strong dual topology now proceeds as in the case of a TVS. Note that if   is a TVS whose continuous dual space separates point on   then   is part of a canonical dual system   where   In the special case when   is a locally convex space, the strong topology on the (continuous) dual space   (that is, on the space of all continuous linear functionals  ) is defined as the strong topology   and it coincides with the topology of uniform convergence on bounded sets in   i.e. with the topology on   generated by the seminorms of the form

 
where   runs over the family of all bounded sets in   The space   with this topology is called strong dual space of the space   and is denoted by  

Definition on a TVS

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Suppose that   is a topological vector space (TVS) over the field   Let   be any fundamental system of bounded sets of  ; that is,   is a family of bounded subsets of   such that every bounded subset of   is a subset of some  ; the set of all bounded subsets of   forms a fundamental system of bounded sets of   A basis of closed neighborhoods of the origin in   is given by the polars:

 
as   ranges over  ). This is a locally convex topology that is given by the set of seminorms on  :   as   ranges over  

If   is normable then so is   and   will in fact be a Banach space. If   is a normed space with norm   then   has a canonical norm (the operator norm) given by  ; the topology that this norm induces on   is identical to the strong dual topology.

Bidual

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The bidual or second dual of a TVS   often denoted by   is the strong dual of the strong dual of  :

 
where   denotes   endowed with the strong dual topology   Unless indicated otherwise, the vector space   is usually assumed to be endowed with the strong dual topology induced on it by   in which case it is called the strong bidual of  ; that is,
 
where the vector space   is endowed with the strong dual topology  

Properties

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Let   be a locally convex TVS.

  • A convex balanced weakly compact subset of   is bounded in  [1]
  • Every weakly bounded subset of   is strongly bounded.[2]
  • If   is a barreled space then  's topology is identical to the strong dual topology   and to the Mackey topology on  
  • If   is a metrizable locally convex space, then the strong dual of   is a bornological space if and only if it is an infrabarreled space, if and only if it is a barreled space.[3]
  • If   is Hausdorff locally convex TVS then   is metrizable if and only if there exists a countable set   of bounded subsets of   such that every bounded subset of   is contained in some element of  [4]
  • If   is locally convex, then this topology is finer than all other  -topologies on   when considering only  's whose sets are subsets of  
  • If   is a bornological space (e.g. metrizable or LF-space) then   is complete.

If   is a barrelled space, then its topology coincides with the strong topology   on   and with the Mackey topology on generated by the pairing  

Examples

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If   is a normed vector space, then its (continuous) dual space   with the strong topology coincides with the Banach dual space  ; that is, with the space   with the topology induced by the operator norm. Conversely  -topology on   is identical to the topology induced by the norm on  

See also

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References

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Bibliography

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  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
  • Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.