The Banach-Steinhaus Theorem in Abstract Duality Pairs


  • Li Ronglu Harbin Institute of Technology.
  • Charles Swartz New Mexico State University.





Let E, F be sets and G a Hausdorff, abelian topological group with b : E X F→ G; we refer to E, F, G as an abstract duality pair with respect to G or an abstract triple and denote this by (E,F : G). Let (Ei,Fi : G) be abstract triples for i = 1, 2. Let Fi be a family of subsets of    Fi    and let    τFi(Ei)    =    τi be    the    topology on    Ei   of uniform convergence on the members of Fi. Let J be a family of mappings from Ei to E2. We consider conditions which guarantee that J is τ12 equicontinuous. We then apply the results to obtain versions of the Banach-Steinhaus Theorem for both abstract triples and for linear operators between locally convex spaces.

Author Biographies

Li Ronglu, Harbin Institute of Technology.

Department of Mathematics.

Charles Swartz, New Mexico State University.

Department of Mathematical Sciences.


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

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

[K2] G. K¨othe, Topological Vector Spaces II, Springer-Verlag, Berlin, (1979).

[LC] R. Li and M. Cho, A Banach-Steinhaus Type Theorem Which is Valid for every Locally Convex Space, Applied Functional Anal., 1, pp. 146- 147, (1993).

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

[Sw2] C. Swartz, The Uniform Boundedness Principle for Arbitrary Locally Convex Spaces, Proy. J. Math., 26, pp. 245-251, (2007).

[Sw3] C. Swartz, Multiplier Convergent Series, World Sci. Publishing, Singapore, (2009).

[Sw4] C. Swartz, Infinite Matrices and the Gliding Hump, World Sci. Publishing, Singapore, (1996).

[Wi1] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, N. Y., (1978).

[Wi2] A. Wilansky, Summability through Functional Analysis, North Holland, Amsterdam, (1984).

How to Cite

L. Ronglu and C. Swartz, “The Banach-Steinhaus Theorem in Abstract Duality Pairs”, Proyecciones (Antofagasta, On line), vol. 34, no. 4, pp. 391-399, 1.