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Graphical convergence of continuous functions

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Abstract

Let X and Y be metrizable spaces. We show that convergence of a net of continuous functions 〈f λ 〉 to a continuous function f in the graph topology for C(X,Y) is equivalent to the uniform convergence of the net of associated distance functionals for the graphs with respect to each compatible metric on X×Y. Remarkably, no weaker convergence results if uniform convergence is replaced by pointwise convergence in the last statement.

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Authors and Affiliations

  1. Department of Mathematics, California State University Los Angeles, 5151 State University Drive, Los Angeles, CA, 90032, USA

    Gerald Beer

  2. 96 Dewson Street, Toronto, Ontario, M6H 1H3, Canada

    Somashekhar Naimpally

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  1. Gerald Beer
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  2. Somashekhar Naimpally
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Correspondence to Gerald Beer.

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Beer, G., Naimpally, S. Graphical convergence of continuous functions. Acta Math Hung 140, 305–315 (2013). https://doi.org/10.1007/s10474-013-0294-z

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  • DOI: https://doi.org/10.1007/s10474-013-0294-z

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