Abstract
If A is (s → s) continuous, and if {f j } is a net w-convergent to ƒ, then (Af j , g) = (f j , A*g) → (f A*g) = (Af, g) for all g, so that Af j → Af (w). This proves that A is (w → w) continuous. Note that the assumption of (s → s) continuity was tacitly, but heavily, used via the existence of the adjoint A*.
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© 1982 Springer-Verlag New York Inc.
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Halmos, P.R. (1982). Properties of Compactness. In: A Hilbert Space Problem Book. Graduate Texts in Mathematics, vol 19. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9330-6_69
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DOI: https://doi.org/10.1007/978-1-4684-9330-6_69
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4684-9332-0
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