A GLIDING HUMP PROPERTY AND
BANACH-MACKEY SPACES

CHARLES SWARTZ

New Mexico State University - U. S. A.

 

Abstract

We consider the Banach-Mackey property for pairs of vector spaces E and E' 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 anoter 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.

 

References

 

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

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

[DFP] S. Díaz, M. Florencio and P. Paúl, A uniform boundedness theorem for for L^¥ (m,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 aplications 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 Vector-Valued 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. Über 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 singuliéres, 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, Über 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 Aplications, 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).

 

Received : December, 2000.

Charles Swartz

Department of Mathematical Sciences

New Mexico State University

Las Cruces, NM 88003

USA

E-mail: cswartz@nmsu.edu