Differential Equations, Fourier Series, and Hilbert Spaces by Raffaele Chiappinelli

Author:Raffaele Chiappinelli
Language: eng
Format: epub
Publisher: De Gruyter
Published: 2023-09-18T07:36:42.442000+00:00

Example 2.7.3.

The linear integral operator T acting in defined in (2.7.12) is compact. This fact rests on a fundamental result in function theory, the Ascoli–Arzelà theorem, which characterizes the relatively compact subsets B of asking that they should be bounded and equicontinuous, meaning that given there exists a such that for every with and every we have

References for the statement and proof of Ascoli–Arzelà’s theorem are for instance Dieudonné [7] and Kolmogorov–Fomin [9]. To show that the set is equicontinuous if is bounded, we use again the uniform continuity of k in to ensure that given , there is a so that (2.7.13) holds. Then the definition (2.7.12) of T shows that if ,

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