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The Theorems of Montel and Vitali

  • Reinhold Remmert
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 172)

Abstract

In infinitesimal calculus, the principle of selection of convergent sequences in bounded subsets M of ℝn is crucial: Every sequence of points in M has a subsequence that converges in ℝn (Bolzano-Weierstrass property). The extension of this accumulation principle to sets of functions is fundamental for many arguments in analysis. But caution is necessary: There are sequences of real-analytic functions from the interval [0, 1] into a fixed bounded interval that have no convergent subsequences. A nontrivial example is the sequence sin 2nπx; cf. 1.1.

Keywords

Holomorphic Function Normal Family Convergent Subsequence Riemann Mapping Theorem Identity Theorem 
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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Reinhold Remmert
    • 1
  1. 1.Mathematisches InstitutWestfälische Wilhelms—Universität MünsterMünsterGermany

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