rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    non sort: The
    title: Real Numbers
    sub title: An Introduction to Set Theory and Analysis
  name:
    name part: Stillwell, John.
    role:
      role term: creator
    role:
      role term: author.
  name:
    name part: SpringerLink (Online service)
  type of resource: text
  origin info:
    place:
      place term: gw
    date issued: 2013
    issuance: monographic
  language:
    language term: eng
  physical description:
    form: electronic
    form: electronic resource
    reformatting quality: access
    form: remote
    extent: XVI, 244 p. 62 illus. online resource.
  abstract: While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics. Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.
  table of contents: The Fundamental Questions -- From Discrete to Continuous -- Infinite Sets -- Functions and Limits -- Open Sets and Continuity -- Ordinals -- The Axiom of Choice -- Borel Sets -- Measure Theory -- Reflections -- Bibliography -- Index.
  note: by John Stillwell.
  subject:
    topic: Mathematics
  subject:
    topic: Functions of real variables
  subject:
    topic: History
  subject:
    topic: Mathematical logic
  subject:
    topic: Mathematics
  subject:
    topic: Real Functions
  subject:
    topic: Mathematical Logic and Foundations
  subject:
    topic: History of Mathematical Sciences
  classification: QA331.5
  classification: 515.8
  related item:
    title info:
      title: Springer eBooks
  related item:
    title info:
      title: Undergraduate Texts in Mathematics
  identifier: 9783319015774
  identifier: http://dx.doi.org/10.1007/978-3-319-01577-4
  record info:
    record creation date: 131016
    record change date: 20151103124113.0
    record identifier: 978-3-319-01577-4