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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Bibliography
BLASCHKE, W.: Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen, Ber. Verh. Königl. Sächs. Ges. Wiss. Leipzig 67, 194–200 (1915); Ges. Werke 6, 187–193.
BURCKEL, R. B.: An Introduction to Classical Complex Analysis, vol. 1, Birkhäuser, 1979.
CARLEMAN, T.: Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen, Ark. för Mat. Astron. Fys. 17, no. 9 (1923).
CARATHÉODORY, C. and E. LANDAU: Beiträge zur Konvergenz von Funktionenfolgen, Sitz. Ber. Königl. Preuss. Akad. Wiss., Phys.-math. Kl. 26, 587–613 (1911); CARATHÉODORY’S Ges. Math. Schriften 3, 13–44; LANDAU’s Coll. Works 4, 349–375.
GOLITSCHEK, M. V.: A short proof of Müntz’s theorem, Journ. Approx. Theory 39, 394–395 (1983).
HERMITE, C. and T.-J. STIELTJES: Correspondance d’Hermite et de Stieltjes, vol. 2, Gauthier-Villars, Paris, 1905.
HILBERT, D.: Über das Dirichletsche Prinzip, Jber. DMV 8, 184–188 (1899); Ges. Abh. 3, 10–14.
JENTZSCH, R.: Untersuchungen zur Theorie der Folgen analytischer Funktionen, Acta Math. 41, 219–251 (1917).
KOEBE, P.: Ueber die Uniformisierung beliebiger analytischer Kurven, Dritte Mitteilung, Nachr. Königl. Ges. Wiss. Göttingen, Math. phys. Kl. 1908, 337–358.
LINDELOF, E.: Démonstration nouvelle d’un théorème fondamental sur les suites de fonctions monogènes, Bull. Soc. Math. France 41, 171–178 (1913).
LÖWNER, K. and T. RadÔ: Bemerkung zu einem Blaschkeschen Konvergenzsatze, Jber. DMV 32, 198–200 (1923).
MONTEL, P.: Sur les suites infinies de fonctions, Ann. Sci. Éc. Norm. Sup. 24, 233–334 (1907).
Mo2] MONTEL, P.: Leçons sur les familles normales de fonctions analytiques et
leurs applications,Gauthier-Villars, Paris, 1927; reissued 1974 by Chelsea Publ. Co., New York.
MÜNTZ, C. H.: Über den Approximationssatz von Weierstraß, Math. Abh. H. A. Schwarz gewidmet, 303–312, Julius Springer, 1914.
OSGOOD, W. F.: Note on the functions defined by infinite series whose terms are analytic functions of a complex variable; with corresponding theorems for definite integrals, Ann. Math.,2nd ser., 3, 25–34 (1901–1902).
PORTER, M. B.: Concerning series of analytic functions, Ann. Math.,2nd ser., 6, 190–192 (1904–1905).
ROGERS, L. C. G.: A simple proof of Müntz’s theorem, Math. Proc. Cambridge Phil. Soc. 90, 1–3 (1981).
RUDIN, W.: Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
RUNGE, C.: Zur Theorie der analytischen Functionen, Acta Math. 6, 245–248 (1885).
STIELTJES, T.-J: Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse 8, 1–22 (1894).
SAKS, S. and A. ZYGMUND: Analytic Functions, 3rd ed., Elsevier, Warsaw, 1971.
VITALI, G.: Sopra le serie de funzioni analitiche, Rend. Ist. Lombardo, 2. Ser. 36, 772–774 (1903) and Ann. Mat. Pur. Appl., 3. Ser. 10, 65–82 (1904).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media New York
About this chapter
Cite this chapter
Remmert, R. (1998). The Theorems of Montel and Vitali. In: Classical Topics in Complex Function Theory. Graduate Texts in Mathematics, vol 172. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2956-6_7
Download citation
DOI: https://doi.org/10.1007/978-1-4757-2956-6_7
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98221-2
Online ISBN: 978-1-4757-2956-6
eBook Packages: Springer Book Archive