A gliding hump property and banach-mackey spaces

  • Charles Swartz New Mexico State University.


We consider the Banach–Mackey property for pairs of vector spaces E and E0 which are in duality. Let A be an algebra of sets and assume that P is an additive map from A into the projection operators on E. We define a continuous gliding hump property for the map P and show that pairs with this gliding hump property and another measure theoretic property are Banach-Mackey pairs, i. e., weakly bounded subsets of E are strongly bounded. Examples of vector valued function spaces, such as the space of Pettis integrable functions, which satisfy these conditions are given.

Biografía del autor/a

Charles Swartz, New Mexico State University.
Department of Mathematical Sciences.


[ B] S. Banach, Oeuvres II, PWN, Warsaw, (1979).

[ BF] J. Boos and D. Fleming, Gliding Hump Properties and Some Applications, Int. J. Math. Math. Sci., 18, pp. 121-132, (1995).

[ DFP] S. Díaz, M. Florencio and P. Paúl, A uniform boundedness theorem for L ∞ (µ, E), Arch. Math. (Basel), 60, pp. 73-78, (1993).

[ DFFP1] S. Díaz, A. Fernández, M. Florencio and P. Paúl, An abstract Banach-Steinhaus theorem and applications to function spaces, Resultate Math., 23, pp. 242-250, (1993).

[ DFFP2] S. Díaz, A. Fernández, M. Florencio and P. Paúl, A Wide Class of Ultrabornological Spaces of Measurable Functions, J. Math. Anal. Appl., 190, pp. 697-713, (1995).

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

[ Do] I. Dobrakov, On Integration in Banach Spaces I, Czech. Math. J., 20, pp. 511-536, (1970).

[ DFP1] L. Drewnowski, M. Florencio, and P. Paúl, The Space of Pettis Integrable Functions is Barrelled, Proc. Amer. Math. Soc., 114, pp. 341-351, (1992).

[ DFP2] L. Drewnowski, M. Florencio and P. Paúl, Uniform boundedness of operators and barrelledness in spaces with Boolean algebras of projections, Atti. Sem. Mat. Fis., Univ. Modena XLI, pp. 317-329, (1993).

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

[ FMP] M. Florencio, F. Mayoral and P. Paúl, Diedonné-Köthe Duality for Vector-Valued Function Spaces, Quaest. Math., 20, pp. 185-214, (1997).

[ FM] D. Fremlin and J. Mendoza, On the Integration of VectorValued Functions, Illinois J. Math., 38, pp. 127-147, (1994).

[ G] R. Gordan, The McShane Integral of Banach-Valued Functions, Illinois J. Math., 34, pp. 557-567, (1990).

[ Ha] H. Hahn, Uber Folgen linearen Operationen, Monatsch. für Math. und Phys., 32, pp. 1-88, (1922).

[ HT] E. Hellinger and O. Toeplitz, Gründlagen für eine Theorie den unendlichen Matrizen, Math. Ann., 69, pp. 289-330, (1910).

[ Hi] T.H. Hilldebrandt, On Uniform Limitedness of Sets of Functional Operations, Bull. Amer. Math. Soc., 29, pp. 309-315, (1923).

[ L] H. Lebesgue, Sur les intégrales sìnguliéres, Ann. de Toulouse, 1, pp. 25-117, (1909).

[ RR] K. P. S. Rao and M. Rao, Theory of Charges, Academic Press, N. Y., (1983).

[ Sch] J. Schur, Uber lineare Transformation in der Theorie die unendlichen Reihen, J. Reine Angew Math., 151, pp. 79-111, (1920).

[ Sw1] C. Swartz, An Introduction to Functional Analysis, Marcel Dekker, N.Y., (1992).

[ Sw2] C. Swartz, Measure, Integration and Functional Spaces, World Sci. Publ., Singapore, (1994).

[ Sw3] C. Swartz, Infinite Matrices and the Gliding Hump, World Sci. Publ., Singapore, (1996).

[ Sw4] C. Swartz, Beppo Levi’s Theorem for the Vector-Valued McShane Integral and Applications, Bull.Belgian Math. Soc., 4, pp. 589-599, (1997).

[ Sw5] C. Swartz, Topological Properties of the Space of Integrable Functions with respect to a Charge, Ricerche di Mat., to appear.

[ Sz] P. Szeptycki, Notes on integral transformations, Diss. Math., 231 (1984).

[ Wi] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, N. Y., (1978).
Cómo citar
Swartz, C. (2017). A gliding hump property and banach-mackey spaces. Proyecciones. Revista De Matemática, 20(2), 243-261. https://doi.org/10.4067/S0716-09172001000200007