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A new and simple proof of Schauder's theorem

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arxiv 1010.1298 v4 pith:ZFURKKKF submitted 2010-10-06 math.FA

classification math.FA
keywords theoremproofschauderadjointanalysisanythingargumentarzela--ascoli
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Schauder's theorem asserts that a bounded linear operator between Banach spaces is compact if ad only if its adjoint is. We give a new proof of this result, which is both short and completely elementary in the sense that it does not depend on anything beyond basic functional analysis, i.e., the Hahn--Banach theorem and some of its consequences; in particular, we avoid the Arzela--Ascoli theorem (and any kind of related diagonal argument).

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  1. Avoiding spectral pollution for transfer operators using residuals

    math.DS 2025-07 conditional novelty 6.0 of 10

    A residual computation for kernelized dynamic mode decomposition gives a necessary condition for eigenvalues of transfer operators, enabling detection of spectral pollution.

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