Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Buchtitelbild

2013 | OriginalPaper | Buchkapitel

1. Introduction

verfasst von : Kenneth A. Ross

Erschienen in: Elementary Analysis

Verlag: Springer New York

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

The underlying space for all the analysis in this book is the set of real numbers. In this chapter we set down some basic properties of this set. These properties will serve as our axioms in the sense that it is possible to derive all the properties of the real numbers using only these axioms. However, we will avoid getting bogged down in this endeavor. Some readers may wish to refer to the appendix on set notation.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Nächstes Kapitel Sequences
Fußnoten
1
Polynomials like this, where the highest power has coefficient 1, are called monic polynomials.
 
2
An integer p ≥ 2 is a prime provided the only positive factors of p are 1 and p.
 
Literatur
[1]
Abbott, S.D.: Understanding Analysis. Springer, New York (2010)
[2]
Bagby, R.J.: Introductory Analysis – A Deeper View of Calculus. Academic, San Diego (2001)MATH
[3]
Bauldry, W.C.: Introduction to Real Analysis – An Educational Approach. Wiley (2010)
[4]
Beals, R.: Advanced Mathematical Analysis. Graduate Texts in Mathematics, vol. 12. Springer, New York/Heidelberg/Berlin (1973). Also, Analysis–an Introduction, Cambridge University Press 2004
[5]
Bear, H.S.: An Introduction to Mathematical Analysis. Academic, San Diego (1997)MATH
[6]
Beardon, A.F.: Limits – A New Approach to Real Analysis. Undergraduate Texts in Mathematics. Springer, New York/Heidelberg/Berlin (1997)MATH
[7]
[8]
Birkhoff, G., Mac Lane, S.: A Survey of Modern Algebra. Macmillan, New York (1953). A. K. Peters/CRC 1998
[9]
Boas, R.P. Jr.: A Primer of Real Functions, 4th edn. Revised and updated by Harold P. Boas. Carus Monograph, vol. 13. Mathematical Association of America, Washington, DC (1996)
[10]
[11]
Botsko, M.W.: Quicky problem. Math. Mag. 85, 229 (2012)
[12]
Brannan, D.: A First Course in Mathematical Analysis. Cambridge University Press, Cambridge/New York (2006)MATHCrossRef
[13]
Bressoud, D.: A Radical Approach to Real Analysis, 2nd edn. The Mathematical Association of America, Washington, DC (2007)MATH
[14]
Burckel, R.B.: An Introduction to Classical Complex Analysis, vol. 1. Birkhäuser, Basel (1979)CrossRef
[15]
[16]
Clark, C.: The Theoretical Side of Calculus. Wadsworth, Belmont (1972). Reprinted by Krieger, New York 1978
[17]
Corominas, E., Sunyer Balaguer, F.: Conditions for an infinitely differentiable function to be a polynomial, Rev. Mat. Hisp.-Amer. (4) 14, 26–43 (1954). (Spanish)
[18]
[19]
Dangello, F., Seyfried, M.: Introductory Real Analysis. Houghton Mifflin, Boston, (2000)
[20]
Donoghue, W.F. Jr.: Distributions and Fourier Transforms. Academic, New York (1969)MATH
[21]
Dunham, W.: The Calculus Gallery: Masterpieces from Newton to Lebesgue. Princeton University Press, Princeton/Woodstock (2008)MATH
[22]
Fitzpatrick, P.M.: Real Analysis. PWS, Boston (1995)
[23]
Gardiner, A.: Infinite Processes, Background to Analysis. Springer, New York/Heidelberg/Berlin (1982). Republished as Understanding Infinity – The Mathematics of Infinite Processes. Dover 2002
[24]
[25]
Gaskill, H.S., Narayanaswami, P.P.: Elements of Real Analysis. Prentice-Hall, Upper Saddle River (1998)MATH
[26]
Gaughan, E.D.: Introduction to Analysis, 5th edn. American Mathematical Society, Providence (2009)MATH
[27]
Gordon, R.A.: Real Analyis – A First Course, 2nd edn. Addison-Wesley, Boston (2002)
[28]
[29]
[30]
[31]
Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Graduate Texts in Mathematics, vol. 25. Springer, New York/Heidelberg/Berlin (1975)
[32]
Hijab, O.: Introduction to Calculus and Classical Analysis. Undergraduate Texts in Mathematics, 2nd edn. Springer, New York/Heidelberg/Berlin (2007)
[33]
Hoffman, K.: Analysis in Euclidean Space. Prentice-Hall, Englewood Cliffs (1975). Republished by Dover 2007
[34]
Johnsonbaugh, R., Pfaffenberger, W.E.: Foundations of Mathematical Analysis. Marcel Dekker, New York (1980). Republished by Dover 2010
[35]
Kantrowitz, R.: Series that converge absolutely but don’t converge. Coll. Math. J. 43, 331–333 (2012)CrossRef
[36]
Kenton, S.: A natural proof of the chain rule. Coll. Math. J. 30, 216–218 (1999)CrossRef
[37]
Kosmala, W.: Advanced Calculus – A Friendly Approach. Prentice-Hall, Upper Saddle River (1999)MATH
[38]
Krüppel, M.: On the zeros of an infinitely often differentiable function and their derivatives. Rostock. Math. Kolloq. 59, 63–70 (2005)MATH
[39]
Landau, E.: Foundations of Analysis. Chelsea, New York (1951). Republished by American Mathematical Society 2001
[40]
Lang, S.: Undergraduate Analysis. Undergraduate Texts in Mathematics, 2nd edn. Springer, New York/Heidelberg/Berlin (2010)
[41]
Lay, S.R.: Analysis – An Introduction to Proof, 4th edn. Prentice-Hall (2004)
[42]
Lewin, J.: A truly elementary approach to the bounded convergence theorem. Amer. Math. Monthly 93, 395–397 (1986)MathSciNetCrossRef
[43]
Lewin, J., Lewin, M.: An Introduction to Mathematical Analysis, 2nd edn. McGraw-Hill, New York (1993)
[44]
[45]
Lynch, M.: A continuous function which is differentiable only at the rationals. Math. Mag. 86, April issue (2013)
[46]
Mattuck, A.: Introduction to Analysis. Prentice-Hall, Upper Saddle River (1999)MATH
[47]
Morgan, F.: Real Analysis. American Mathematical Society, Providence (2005)
[48]
[49]
Niven, I.: Irrational Numbers. Carus Monograph, vol. 11. Mathematical Association of America, Washington, DC (1956)
[50]
Niven, I., Zuckerman, H.S., Montgomery, H.I.: An Introduction to the Theory of Numbers, 5th edn. Wiley, New York (1991)
[51]
Pedrick, G.: A First Course in Analysis. Undergraduate Texts in Mathematics. Springer, New York/Heidelberg/Berlin (1994)MATHCrossRef
[52]
Phillips, E.: An Introduction to Analysis and Integration Theory. Intext Educational Publishers, Scranton/Toronto/London (1971)MATH
[53]
Protter, M.H., Morrey, C.B.: A First Course in Real Analysis. Undergraduate Texts in Mathematics, 2nd edn. Springer, New York/Heidelberg/Berlin (1997)
[54]
[55]
Randolph, J.F.: Basic Real and Abstract Analysis. Academic, New York (1968)MATH
[56]
Reed, M.: Fundamental Ideas of Analysis. Wiley, New York (1998)MATH
[57]
Robdera, M.A.: A Concise Approach to Mathematical Analysis. Springer, London/New York (2003)MATHCrossRef
[58]
Rosenlicht, M.: Introduction to Analysis. Dover, New York (1985)
[59]
Ross, K.A.: First digits of squares and cubes. Math. Mag. 85, 36–42 (2012)MATH
[60]
Ross, K.A., Wright, C.R.B.: Discrete Mathematics, 5th edn. Prentice-Hall, Upper Saddle River (2003)
[61]
Rotman, J.: Journey into Mathematics – An Introduction to Proofs. Prentice-Hall, Upper Saddle River (1998)MATH
[62]
Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976)MATH
[63]
Schramm, M.J.: Introduction to Real Analysis. Prentice-Hall, Upper Saddle River (1996). Dover 2008
[64]
[65]
Stromberg, K.: An Introduction to Classical Real Analysis. Prindle, Weber & Schmidt, Boston (1980)
[66]
Thim, J.: Continuous Nowhere Differentiable Functions, Master’s thesis (2003), Luleå University of Technology (Sweden). http://​epubl.​luth.​se/​1402-1617/​2003/​320/​LTU-EX-03320-SE.​pdf
[67]
[68]
[69]
Weierstrass, K.: Über continuirliche Funktionen eines reellen Arguments, die für keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen, Gelesen Akad. Wiss. 18 July 1872, and J. für Mathematik 79, 21–37 (1875)
[70]
Wolf, R.S.: Proof, Logic, and Conjecture: The Mathematician’s Toolbox. W. H. Freeman, New York (1998)
[71]
[72]
Zhou, L., Markov, L.: Recurrent proofs of the irrationality of certain trigonometric values. Amer. Math. Monthly 117, 360–362 (2010)MathSciNetMATHCrossRef
[73]
Zorn, P.: Understanding Real Analysis. A. K. Peters, Natick (2010)MATH
Metadaten
Titel
Introduction
verfasst von
Kenneth A. Ross
Copyright-Jahr
2013
Verlag
Springer New York
Buch
Elementary Analysis

Print ISBN: 978-1-4614-6270-5

Electronic ISBN: 978-1-4614-6271-2

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-1-4614-6271-2_1