Gliding Hump Properties in Abstract Duality Pairs with Projections


  • Charles Swartz New Mexico State University.



Topological vector spaces, bounded sets, convergent series, espacios vectoriales topológicos, conjuntos acotados, series convergentes.


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.

Author Biography

Charles Swartz, New Mexico State University.

Mathematics Department.


[1] R. Bartle, A general bilinear vector integral, Studia Math., 15, pp. 337-352, (1956).

[2] C. Bosch and C. Swartz, Functional Calculi, World Sci. Publ., Singapore, (2013).

[3] C. Cho, R. Li and C. Swartz, Subseries convergence in abstract duality pairs, Proy. J. Math., 33, pp. 447-470, (2014).

[4] J. Diestel and J. Uhl, Vector Measures, Amer. Math. Soc. Surveys #15, Providence, (1977).

[5] L. Drewnowski, M. Florencio and P. Paul, The Space of Pettis Integrable Functions is Barrelled, Proc, Amer. Math. Soc., 114, pp. 687-694, (1992).

[6] N. Dunford and J. Schwartz, Linear Operators I, Interscience, N. Y., (1958).

[7] P. K. Kamthan and M. Gupta, Sequence Spaces and Series, Marcel Dekker, N. Y., (1981).

[8] I. Kluvanek and G. Knowles, Vector Measures and Control Systems, North-Holland, Amsterdam, (1976).

[9] Lee Peng Yee, Sequence Spaces and the Gliding Hump Property, Southeast Asia Bull. Math., Special Issue, pp. 65-72, (1993).

[10] Li Ronglu and C. Swartz, Spaces for Which the Uniform Boundedness Principle Holds, Studia Sci. Math. Hung., 27, pp. 379-384, (1992).

[11] K.Musial, Topics in the Theorey of Pettis Integration, Rend. Instituto Mat. Univ. Trieste, Vol. XXIII, (1991).

[12] D. Noll and W. Stadler, Abstract sliding hump techniques and characterizations of barrelled spaces, Studia Math., 94, pp. 103-120, (1989).

[13] T. V. Panchapagesan, The Bartle-Dunford-Schwartz Integral, Birkhauser, Basel, (2008).

[14] C. Stuart, Weak Sequential Completeness of -duals in Sequence Spaces, Rocky Mount. Math. J., 26, pp. 1559-1568, (1996).

[15] C. Swartz, Infinite Matrices and the Gliding Hump, World Sci. Publ., Singapore, (1996).

[16] C. Swartz, Multiplier Convergent Series. World. Sci. Publ., Singapore, (2009).

[17] C. Swartz, Measure, Integration and Function Spaces, World Sci. Publ., Singapore, (1994).

[18] A. Wilansky, Modern Methods in Topological Vector Spaces, McGrawHill, N. Y., (1978).

[19] Junde Wu, Jianwen Luo and Chengri Cui, The Abstract Gliding Hump Properties and Applications, Taiwan. J. Math., 10, pp. 639-649, (2006).

[20] Zheng Fu, Cui Chengri and Li Ronglu, Abstract Gliding Hump Properties in the Vector-Valued Dual Pair, Acta Anal. Funct. Appl., 12, pp. 322-327, (2010).



How to Cite

C. Swartz, “Gliding Hump Properties in Abstract Duality Pairs with Projections”, Proyecciones (Antofagasta, On line), vol. 35, no. 3, pp. 339-367, Mar. 2017.