The Banach-Steinhaus Theorem in Abstract Duality Pairs
DOI:
https://doi.org/10.4067/S0716-09172015000400007Keywords:
Matemáticas.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) = 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 τ1-τ2 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.References
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[Wi2] A. Wilansky, Summability through Functional Analysis, North Holland, Amsterdam, (1984).
[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
[1]
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.
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