Introduction to Real Analysis

Erdoğan S. Şuhubi
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Erdoğan S. Şuhubi: Yeditepe University, Department of Mathematics

Chapter Chapter III in Functional Analysis, 2003, pp 157-220 from Springer

Abstract: Abstract The set of real numbers, which possesses quite a rich structure because it is both a field and, furthermore, a linear vector space with respect to operations of addition and multiplication in this field, has many interesting properties which are in complete harmony with our intuitive expectations. Hence, we are easily ready to accept and exploit these properties as if they were originated from natural processes. Extensive studies and interpretations of these properties during many centuries have led to remarkable developments of certain level of abstraction such as the concepts of limit and continuity which highly transcend our intuitive understanding. This rather abstract approach, starting in the early 17th century, helped shape the mathematical analysis and gave rise to tremendous accomplishments both in extending the fundamental concepts and methods on the on hand and in effectiveness of implementation of mathematics in better understanding and controlling the physical world on the other. Almost all the advances made in the analysis up to the beginning of the 20th century were accomplished by directly extending or generalising various properties observed in the set of real numbers.

Keywords: Rational Number; Open Interval; Open Cover; Cauchy Sequence; Accumulation Point (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-017-0141-9_3

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DOI: 10.1007/978-94-017-0141-9_3

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