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Download Fundamentals of Abstract Analysis fb2

by Andrew Gleason
Download Fundamentals of Abstract Analysis fb2
Mathematics
  • Author:
    Andrew Gleason
  • ISBN:
    0867202092
  • ISBN13:
    978-0867202090
  • Genre:
  • Publisher:
    A K Peters/CRC Press; First Edition edition (November 2, 1992)
  • Pages:
    416 pages
  • Subcategory:
    Mathematics
  • Language:
  • FB2 format
    1583 kb
  • ePUB format
    1725 kb
  • DJVU format
    1434 kb
  • Rating:
    4.6
  • Votes:
    495
  • Formats:
    mobi txt lit lrf


Fundamentals of Abstract Analysis was our prescribed text in my Advanced Calculus class. I got hold of this book from our college library. The way Gleason presented topics form this course was brilliant.

Fundamentals of Abstract Analysis was our prescribed text in my Advanced Calculus class. Although this course is not a basic calculus course, this is a must for mathematics majors who wish to pursue masters and doctorate degrees in the future. This book is recommended for students with a more mathematical maturity than that afforded by the usual freshman-sophomore courses in calculus.

This classic is an ideal introduction for students into the methodology and thinking of higher mathematics. It covers material not usually taught in the more introductory classes and will give students a well-rounded foundation for future studies. Categories: Mathematics\Analysis.

Fundamentals of Abstract Analysis book. This classic is an ideal introduction for students into.

Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them

Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. 1. An Introduction to Linear Algebra.

ANDREW M. GLEASON, Harvard University. Fundamentals of Abstract Analysis. Laddison-wesley, publishing company. Reading, massachusetts palo alto london don mills, ontario contents. Chapter 1. Sets 1-1. The notion of set 1 1-2. Equality 2 1-3. Parentheses 3 1-4.

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Fundamental analysis helps both the novice and the experienced traders, to make, better and . The book has a good introduction to fundamental analysis and it also provides the reader with detailed information on investing

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Rich Vulture
Note, first: Solutions to most Exercises concludes this fine (1991 edition) textbook---spanning more than fifty pages--thus, a useful reference !
" Exercises are arranged roughly in order of increasing order of difficulty." (Preface). Excellent pedagogy !
And, if said solution is unavailable, a hint is provided. Also, reading the preface: "...intended for...fourth-year students, give, or take a year."
Be that as it may, I would not hand this to a third-year student, it would likely be most challenging--too-- for the fourth-year student. In fact, it seems to me that many an instructor would eschew the text primarily due to its idiosyncratic and creative discourse (no integration theory, for instance).
Hopefully, I am in error on that score ! My observations upon this insightful textbook:
(1) Sets, Logic, Functions and Relations--fundamental concepts for which you will spend the initial fifty pages. Abstract, yet necessary foundation.
(2) We note: " an ordered pair will be called a configuration." (page 55). Read: "set-theoretic descriptions are often clumsy, oblique, and non-intuitive.
Still, they are always precise." (page 61). Style of the author: glance at the proof of proposition #5.1.3 (page 66)--short, but sweet !
(3) Chapter Six is where we really hit the ground running: Order..."is an abstraction of the notion of dominance." Careful reading of this chapter will serve one well. Next, a favorite topic of mine:
(4) Mathematical Induction. Approximately ten pages of exposition. The concept of "chains" is introduced. (" a configuration of a set and a function.")
(5) We read: "The structure we call the real number system is to be an idealization of the numbers we use to measure quantity in the physical world."
This, the eighth chapter on Fields. Of note, the proof (page 109) which "typifies the use of fixed-point theorems." Real Numbers "constructed"
in chapter nine where "it provides an excellent demonstration of set-theoretic techniques." (pages 112-128).
(6) Brief interlude, of six pages--introduces complex numbers. More about them later in the textbook (chapter fifteen). Another pedagogic highlight of this book is the interesting discussion of "another form of induction" (see Section 11-4, pages 144-148) which segues to "the axiom of choice."
(7) Chapter Twelve, Limits. We begin with sequences ( both real and complex numbers). A nice exposition of limits and arithmetic intervenes (with an equally nice proof--pages 170 and 171--of proposition #12-2.2. We read: "Most of the skill in finding mathematical proofs consists of choosing appropriate specializations of universally quantified propositions." (page 169). And, we "develop criteria for the existence of limits... these are consequences of the completeness of the real numbers." (page 180). Bolzano-Weierstrauss theorem concludes the chapter.
(8) Next, Infinite Series (page 191): "Most classical books define an infinite series to be a notation, not a mathematical object." and "the distinction between an infinite series and its sum is important, so it seems worthwhile to distinguish them in the notation." Excellent pedagogy !
We get an excellent discussion of the various "tests" for convergence, even a discussion of infinite products (page 215)--brief, yet lucid. We read:
"the situation becomes transparent with the aid of logarithms....since log is an order-preserving bijection from the positive reals to R..."
(9) Topology of Metric Spaces, next. The discussion segues to Continuous Functions (page 239) on to uniform continuity (page 245). Excellent.
Homeomorphism, defined. (page 249). Compact Spaces "resemble in many ways the properties of finite sets." Learn of them, page 266.
(10) Chapter fifteen will reintroduce Complex Functions: Sine, Cosine and Exponential--via Taylor(power) series.(pages 308-315). While there is no integration presented, that which is here is very well presented. In particular thirteen pages of geometric applications concludes the chapter.
That is, "we shall prove some theorems which connect the theory of analytic functions to the geometry of the plane." (page 326).
Summarizing my review: Well written account of fundamental principles generally applied to the term " mathematical analysis."
The exposition is lucid, the proofs are clear, the exercises are both straightforward and challenging.
A minor quibble would be no references or bibliography supplied. The subject index (seven pages) is adequate.
Keep in mind, this is not "re-doing Calculus" at a more abstract vantage--you look elsewhere for that.
Highly recommended, especially as a reference work and a source of inspiration.
Nilarius
Recommended for mathematicians who really want to have a better understanding of the foundations of mathematics! Magnificent work!
Gogal
Fundamentals of Abstract Analysis was our prescribed text in my Advanced Calculus class. I got hold of this book from our college library. The way Gleason presented topics form this course was brilliant. Although this course is not a basic calculus course, this is a must for mathematics majors who wish to pursue masters and doctorate degrees in the future. This book is recommended for students with a more mathematical maturity than that afforded by the usual freshman-sophomore courses in calculus. And one excellent way to gain such maturity is through a beginning course in linear algebra, although it is not a pre-requisite for the book, but of an advantage. This book tackles the "structure" of calculus, wherein theorems are presented in much greater depths and goes "behind the scenes" of the theorems in calculus. I recommend this book for undergraduate math majors and graduates as well.