Gliding Hump Properties in Abstract Duality Pairs with Projections

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 : 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 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.