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The Banach-Steinhaus Theorem for Ordered Spaces

In: Generalized Functions, Convergence Structures, and Their Applications

Author

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  • Charles Swartz

    (New Mexico State University, Department of Mathematical Sciences)

Abstract

Let X and Y be vector lattices and Ti: X → Y a sequence of linear operators which are sequentially continuous with respect to relative uniform convergence. If {Tjx} is relatively uniformly convergent to Tx for each x ∈ X, under appropriate assumptions on the spaces, we show that the linear operator T is also continuous and that the {Ti} are order equicontinuous in a certain sense. We also establish an order version of the Uniform Boundedness Principle.

Suggested Citation

  • Charles Swartz, 1988. "The Banach-Steinhaus Theorem for Ordered Spaces," Springer Books, in: Bogoljub Stanković & Endre Pap & Stevan Pilipović & Vasilij S. Vladimirov (ed.), Generalized Functions, Convergence Structures, and Their Applications, pages 425-432, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-1055-6_45
    DOI: 10.1007/978-1-4613-1055-6_45
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