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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.35 no.3 Antofagasta Sept. 2016

http://dx.doi.org/10.4067/S0716-09172016000300009 

 

Gliding Hump Properties in Abstract Duality Pairs with Projections

 

Charles Swartz

New Mexico State University, U.S.A.


ABSTRACT

Let E, G be Hausdorff topological vector spaces and let F be a vector space. Assume there is a bilinear operator <.,.> : E X F →G such that <.,y> : E →G is continuous for every y £ F. The triple E, F, G is called an abstract duality pair with respect to G or an abstract triple and is denoted by (E,F : G). If {Pj} is a sequence of continuous projections on E, then (E,F : G) is called an abstract triple with projections. Under appropriate gliding hump assumptions, a uniform bounded principle is established for bounded subsets ofE and pointwise bounded subsets of F. Under additional gliding hump assumptions, uniform convergent results are established for series ∑ j=1 < Pjx,y> when x varies over certain subsets of E and y varies over certain subsets of F. These results are used to establish uniform countable additivity results for bounded sets of indefinite vector valued integrals and bounded subsets of vector valued measures.


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Received : April 2016. Accepted : August 2016

 

Charles Swartz
Mathematics Department
New Mexico State University
 Las Cruces, NM 88003,

U. S. A.
e-mail : cswartz@nmsu.edu

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