An abstract gliding hump property


  • Charles Swartz New Mexico State University.



A new definition of almost fuzzy compactness is introduced in Ltopological spaces by means of open L-sets and their inequality when L is a complete DeMorgan algebra. It can also be characterized by closed L-sets, regularly closed L-sets, regularly open L-sets and their inequalities. When L is a completely distributive DeMorgan algebra, its many characterizations are presented.

Author Biography

Charles Swartz, New Mexico State University.

Department of Mathematics.


[Ap] T. Apostol, Mathematical Analysis, Addison-Wesley, Reading, 1975.

[BP] C. Bessaga and A. Pelczynski, On Bases and Unconditional Convergence of Series in Banach Spaces, Studia Math., 17 (1958),151-164.

[Bo] J. Boos, Classical and Modern Methods in Summability, Oxford University Press, Oxford ,2000.

[BSS] J. Boos, C. Stuart and C. Swartz, Gliding hump properties of matrix domains, Analysis Math., 30 (2004), 243-257.

[DU] J. Diestel and J. Uhl, Vector Measures, Amer. Math. Soc., Providence, 1977.

[Di] N. Dinculeanu, Weak Compactness and Uniform Convergence of Operators in Spaces of Bochner Integrable Functions, J. Math. Anal. Appl., 109 (1985), 372-387.

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

[E] R. E. Edwards, Functional Analysis, Holt-Rinehart-Winston, N.Y., 1965.

[FL] W. Filter and I. Labuda, Essays on the Orlicz-Pettis Theorem I, Real. Anal. Exchange, 16 (1990/91), 393-403.

[Ka] N. Kalton, The Orlicz-Pettis Theorem, Contemporary Math., Amer. Math. Soc., Providence, 1980.

[KG] K.G. Kamthan and M. Gupta, Sequence Spaces and Series, Marcel Dekker, N.Y.,1981.

[Kö] G. Köthe, Topological Vector Spaces, Springer-Verlag, Berlin, 1969.

[LW] E. Lacey and R. Whitley, Conditions under which all the Bounded Linear Maps are Compact, Math. Ann., 158 (1965), 1-5, (1965).

[LS] Li, Ronglu and C. Swartz, An Abstract Orlicz-Pettis Theorem and Applications, Proy. J. Math., 27, pp. 155-169, (2008).

[LWZ] Li, Ronglu and Wang, Fubin and Zhong, S., The strongest intrinsic meaning of sequential-evaluation convergence, Topology and its Appl., 154, pp. 1195-1205, (2007).

[MR] C. W. McArthur and J. Retherford, Some Applications of an Inequality in Locally Convex Spaces, Trans. Amer. Math. Soc., 137, pp. 115-123; (1969).

[MM] G. Metafune and V. B. Moscatelli, On the Space lp+ = ∩q>pl q, Math. Nachr., 147, pp. 7-12, (1990).

[No] D. Noll, Sequential Completeness and Spaces with the Gliding Hump Property, Manuscripta Math., 66, pp. 237-252, (1990).

[Or] W. Orlicz, Beitrage zur Theorie der Orthogalent wichlungen II, Studia Math., 1, pp. 241-255, (1929).

[Pe] B. J. Pettis, Integration in Vector Spaces, Trans. Amer. Math. Soc. 44, pp. 277-304, (1938).

[St] C. Stuart, Weak Sequential Completeness of ß-duals, Rocky Mountain Math. J., 26, pp. 1559-1568, (1996).

[StSw1] C. Stuart and C. Swartz, Uniform Convergence in the Dual of a Vector-Valued Sequence Space, Taiwanese J. Math., 7, pp. 665-676, (2003).

[StSw2] C. Stuart and C.Swartz, Generalizations of the Orlicz-Pettis Theorem, Proy. J. Math., 24, pp. 37-48, (2005).

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

[Sw2] C. Swartz, Infinite Matrices and the Gliding Hump, World Sci. Publ. Singapore, (1996).

[Sw3] C. Swartz, The Schur and Hahn Theorems for Operator Matrices, Rocky Mountain Math. J., 15, pp. 61-73, (1985).

[Sw4] C. Swartz, A Multiplier Gliding Hump Property for Sequence Spaces, Proy. J. Math., 20, pp. 19-31, (2001).

[Sw5] C. Swartz, Orlicz-Pettis Theorems for Multiplier Convergent Operator Valued Series, Proy. J. Math., 23, pp. 61-72, (2004).

[Sw6] C. Swartz, Uniform Convergence of Multiplier Convergent Series, Proy J. Math., 26, pp. 27-35, (2007).

[Sw7] C. Swartz, Interchanging Orders of Summation for Multiplier Convergent Series, Bol. Soc. Mat. Mexicana (3) 8, pp. 31-35, (2002).

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

[ZLY] hong, S. and Li, Ronglu and Yang, Hong, Summability Results for Matrices of Quasi-Homogeneous Operators, Proy. J. Math., 27, pp. 249-258, (2008).

How to Cite

C. Swartz, “An abstract gliding hump property”, Proyecciones (Antofagasta, On line), vol. 28, no. 1, pp. 89-109, 1.