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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.34 no.4 Antofagasta dic. 2015

http://dx.doi.org/10.4067/S0716-09172015000400007 

Proyecciones Journal of Mathematics Vol. 34, No 4, pp. 391-399, December 2015. Universidad Católica del Norte Antofagasta - Chile

The Banach-Steinhaus Theorem in Abstract Duality Pairs

Min-Hyung Cho

Kum-Oh National Institute of Technology

Korea

Li Ronglu

Harbin Institute of Technology

P. R. C.

Charles Swartz

New Mexico State University

U. S. A.


ABSTRACT

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)    =    τibe    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 τ12equicontinuous. 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.


REFERENCES

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[LC] R. Li and M. Cho, A Banach-Steinhaus Type Theorem Which is Valid for every Locally Convex Space, Applied Functional Anal., 1, pp. 146147, (1993).         [ Links ]

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

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

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

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

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

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

Min-Hyung Cho

Department of Applied Mathematics Kum-Oh National Institute of Technology Kumi,

Korea

e-mail : mignon@kumoh.ac.kr

Li Ronglu (he is deceased) Department of Mathematics Harbin Institute of Technology Harbin,

P. R. C.

and

Charles Swartz

Department of Mathematical Sciences New Mexico State University Las Cruces, NM, 88003,

U. S. A.

e-mail : cswartz@nmsu.edu

Received : October 2015. Accepted : November 2015

This paper was supported by Research Fund, Kumoh National Institute of Technology

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