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Differential- Und Integralrechnung I: Funktionen Einer Reellen Veränderlichen by Hans Grauert, Ingo Lieb

Differential- Und Integralrechnung I: Funktionen Einer Reellen Veränderlichen

by Hans Grauert , Ingo Lieb
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Book cover type: Paperback
  • ISBN13: 9783540075745
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Springer
  • Publisher Imprint: Springer
  • Publication Date:
  • Pages: 206
  • Original Price: GBP 39.99
  • Language: German
  • Edition: 4., Verb. Aufl.
  • Item Weight: 232 grams
  • BISAC Subject(s): Mathematical Analysis

lesungen gemaB solI auch das Buch einem Leser, der keine Vorkenntnisse in hoherer Mathematik besitzt, die Gelegenheit geben, einen moglichst strengen und systematischen Aufbau der Theorie der reellen Funktionen kennenzulernen. Dementsprechend sind aIle Beweise bis in die Einzel- heiten hinein ausgeflihrt, und in den ersten Paragraphen werden wich- tige Beweismethoden eigens erlautert. Dabei nehmen wir jedoch den logischen und mengentheoretischen Gesetzen gegenliber einen naiven", d. h. nicht-axiomatischen, Standpunkt ein. Das gilt besonders flir das Prinzip der vollstandigen Induktion und damit auch flir den Begriff der natlirlichen Zahl und der Folge. Wir geben eine Obersicht iiber den Inhalt des Buches. Grundlegend ist der Begriff der reellen Zahl. 1m ersten Kapitel werden die Axiome des rellen Zahlkorpers mit ihren einfachsten Folge- rungen ausflihrlich besprochen; die unendlich fernen Punkte + 00 und - 00 werden axiomatisch miteingeflihrt. Die nachsten beiden Kapitel sind dem Umgebungsbegriff und dem darauf fuBenden Grenzwertbegriff flir Folgen und Reihen gewidmet. Da wir flir die Definition der Konvergenz die natlirliche (uniforme) Topologie der Zahlengeraden zugrundelegen, bleibt die Konvergenz gegen 00 ausgeschlossen. - Die Begriffe limes superior" und limes inferior" sind so gefaBt, daB sie mit der Definition der halbstetigen Funktionen harnionieren. Reelle Funktionen werden im vierten Kapitel behandelt. Vor den stetigen werden halbstetige Funktionen definiert. Dieser Funktionstyp ist in Kapitel VII flir die Definition von Umgebungen im Funktions- raum wichtig und damit zur Einflihrung des Lebesgueschen Integrals, das in diesem Buch -das unbefriedigende Riemannsche Integral ablOst.

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