Convergence of functions: equi-semicontinuity

Dolecki, Szymon; Salinetti, Gabriella; Wets, Roger J.-B.

doi:10.1090/S0002-9947-1983-0684518-7

Convergence of functions: equi-semicontinuity

Authors: Szymon Dolecki, Gabriella Salinetti and Roger J.-B. Wets
Journal: Trans. Amer. Math. Soc. 276 (1983), 409-430
MSC: Primary 58E30; Secondary 49D99, 54A20
DOI: https://doi.org/10.1090/S0002-9947-1983-0684518-7
MathSciNet review: 684518
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Abstract: We study the relationship between various types of convergence for extended real-valued functionals engendered by the associated convergence of their epigraphs; pointwise convergence being treated as a special case. A condition of equi-semicontinuity is introduced and shown to be necessary and sufficient to allow the passage from one type of convergence to another. A number of compactness criteria are obtained for families of semicontinuous functions; in the process we give a new derivation of the Arzelá-Ascoli Theorem.

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