This is a very carefully written introduction to real analysis. When Apostol published the first edition in 1957, he intended it to be intermediate between calculus and real variables theory, and it still has a strong feeling of being a transitional course. It starts out with several chapters on the number line and point set topology, then proves all the basic facts that are taken for granted in differential calculus courses. It then proceeds into what is new material for most students, with two new theories of integration (Riemann-Stieltjes and Lebesgue), multivariable and vector calculus (focusing on existence theorems such as the Implicit Function Theorem rather than physical applications), some advanced theorems in sequences and series, approximation by sequences of functions and orthonormal bases, and a brief introduction to complex variables.
The book crams a lot of material into a modest number of pages, though without feeling rushed. This is done primarily by sticking to the main threads of the subject in the exposition, while moving a lot of related and more specialized results to the exercises. The proofs themselves are not overly brief, although there’s not much handholding and only a few examples are given, so (a) you have to pay attention, and (b) it’s very helpful to have already taken advanced calculus so that you can orient yourself. In many ways this book resembles the British analysis books of the early twentieth century, such as Hardy’s A Course of Pure Mathematics and Titchmarsh’s Theory of Functions. The approach is generally more modern, and there are many more exercises, but it has the same kind of concision.
The book makes a good balance between simplicity and generality. For example, the Riemann-Stieltjes integral rather than the plain Riemann integral is used for the elementary integration (before Lebesgue). It’s not any harder, introduces some other valuable concepts such as functions of bounded variation, and gives us a tool which is often useful in discrete and discontinuous problems. For another example, Fourier series are developed first for orthogonal systems and then specialized for trigonometric series. Again, this is not any harder and gives us a better insight into why Fourier series work.
Rudin’s Principles of Mathematical Analysis is the one to beat in this field. Apostol’s treatment is not that different from Rudin’s. The books were written about the same time, with Rudin having editions in 1953, 1964, and 1976, and Apostol in 1957 and 1974. The coverage of the two books is roughly similar. Rudin is slightly more abstract and slanted more toward multivariable analysis. Both have concise proofs, a shortage of examples, and numerous challenging exercises. Both cover the Riemann-Stieltjes integral rather than the plain Riemann integral. Both cover the Lebesgue integral, although Rudin is more skimpy and uses the conventional measure theory approach, while Apostol follows Riesz & Sz.-Nagy et al., using a functional-analysis approach through step functions.
Apostol has taken care to modularize his book, so that it can be used for several different courses and the material studied in different orders. Rudin’s treatment is more tightly integrated. This often makes Apostol easier to use as a reference, because everything you need to understand a theorem will be close by.
Author(s): Tom M. Apostol
Publisher: Pearson
Year: 1973
Language: English
Pages: 506
City: Pasadena
Tags: Mathematical Analysis, Advanced Calculus
Chapter 1 The Real and Complex Number Systems
1.1 Introduction
1.2 The field axioms
1.3 The order axioms
1.4 Geometric representation of real numbers
1.5 Intervals
1.6 Integers
1.7 The unique factorization theorem for integers
1.8 Rational numbers
1.9 Irrational numbers
1.10 Upper bounds, maximum element, least upper bound (supremum)
1.11 The completeness axiom
1.12 Some properties of the supremum
1.13 Properties of the integers deduced from the completeness axiom
1.14 The Archimedean property of the real-number system
1.15 Rational numbers with finite decimal representation
1.16 Finite decimal approximations to real numbers
1.17 Infinite decimal representation of real numbers
1.18 Absolute values and the triangle inequality
1.19 The Cauchy-Schwarz inequality
1.20 Plus and minus infinity and the extended real number system R*
1.21 Complex numbers
1.22 Geometric representation of complex numbers
1.23 The imaginary unit
1.24 Absolute value of a complex number
1.25 Impossibility of ordering the complex numbers
1.26 Complex exponentials
1.27 Further properties. of complex exponentials
1.28 The argument of a complex number
1.29 Integral powers and roots of complex numbers
1.30 Complex logarithms
1.31 Complex powers
1.32 Complex sines and cosines
1.33 Infinity and the extended complex plane C*
Exercises
Binder1.pdf
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3
Chapter 2 Some Basic Notions of Set Theory
2.1 Introduction
2.2 Notations
2.3 Ordered pairs
2.4 Cartesian product of two sets
2.5 Relations and functions
2.6 Further terminology concerning functions
2.7 One-to-one functions and inverses
2.8 Composite functions
2.9 Sequences
2.10 Similar (equinumerous) sets
2.11 Finite and infinite sets
2.12 Countable and uncountable sets
2.13 Uncountability of the real-number system
2.14 Set algebra
2.15 Countable collections of countable sets
Exercises
Chapter 3 Elements of Point Set Topology
3.1 Introduction
3.2 Euclidean space R"
3.3 Open balls and open sets in R"
3.4 The structure of open sets in R1
3.5 Closed sets
3.6 Adherent points. Accumulation points
3.7 Closed sets and adherent points
3.8 The Bolzano-Weierstrass theorem
3.9 The Cantor intersection theorem
3.10 The LindelSf covering theorem
3.11 The Heine-Borel covering theorem
3.12 Compactness in R"
3.13 Metric spaces
3.14 Point set topology in metric spaces
3.15 Compact subsets of a metric space
3.16 Boundary of a set
Exercises
Chapter 4 Limits and Continuity
4.1 Introduction
4.2 Convergent sequences in a metric space
4.3 Cauchy sequences
4.4 Complete metric spaces
4.5 Limit of a function
4.6 Limits of complex-valued functions
4.7 Limits of vector-valued functions
4.8 Continuous functions
4.9 Continuity of composite functions.
4.10 Continuous complex-valued and vector-valued functions
4.11 Examples of continuous functions
4.12 Continuity and inverse images of open or closed sets
4.13 Functions continuous on compact sets
4.14 Topological mappings (homeomorphisms)
4.15 Bolzano's theorem
4.16 Connectedness
4.17 Components of a metric space
4.18 Arcwise connectedness
4.19 Uniform continuity
4.20 Uniform continuity and compact sets
4.21 Fixed-point theorem for contractions
4.22 Discontinuities of real-valued functions
4.23 Monotonic functions
Exercises
Chapter 5 Derivatives
5.1 Introduction
5.2 Definition of derivative
5.3 Derivatives and continuity
5.4 Algebra of derivatives
5.5 The chain rule
5.6 One-sided derivatives and infinite derivatives
5.7 Functions with nonzero derivative
5.8 Zero derivatives and local extrema
5.9 Rolle's theorem
5.10 The Mean-Value Theorem for derivatives
5.11 Intermediate-value theorem for derivatives
5.12 Taylor's formula with remainder
5.13 Derivatives of vector-valued functions
5.14 Partial derivatives
5.15 Differentiation of functions of a complex variable
5.16 The Cauchy-Riemann equations
Exercises
Chapter 6 Functions of Bounded Variation and Rectifiable Curves
6.6 Total variation on [a, x] as a function of x
6.7 Functions of bounded variation expressed as the difference of increasing functions
6.8 Continuous functions of bounded variation
6.9 Curves and paths
6.10 Rectifiable paths and arc length
6.11 Additive and continuity properties of arc length
6.12 Equivalence of paths. Change of parameter
Exercises
Chapter 7 The Riemann-Stieltjes Integral
7.1 Introduction
7.2 Notation
7.3 The definition of the Riemann-Stieltjes integral
7.4 Linear properties
7.5 Integration by parts
7.6 Change of variable in a Riemann-Stieltjes integral
7.7 Reduction to a Riemann integral
7.8 Step functions as integrators
7.9 Reduction of a Riemann-Stieltjes integral to a finite sum
7.10 Euler's summation formula
7.11 Monotonically increasing integrators. Upper and lower integrals
7.12 Additive and linearity properties of upper and lower integrals
7.13 Riemann's condition
7.14 Comparison theorems
7.15 Integrators of bounded variation
7.16 Sufficient conditions for existence of Riemann-Stieltjes integrals
7.17 Necessary conditions for existence of Riemann-Stieltjes integrals
7.18 Mean Value Theorems for Riemann-Stieltjes integrals
7.19 The integral as a function of the interval
7.20 Second fundamental theorem of integral calculus
7.21 Change of variable in a Riemann integral
7.22 Second Mean-Value Theorem for Riemann integrals
7.23 Riemann-Stieltjes integrals depending on a parameter
7.24 Differentiation under the integral sign
7.25 Interchanging the order of integration
7.26 Lebesgue's criterion for existence of Riemann integrals
7.27 Complex-valued Riemann-Stieltjes integrals
Exercises
Chapter 8 Infinite Series and Infinite Products
8.1 Introduction
8.2 Convergent and divergent sequences of complex numbers
8.3 Limit superior and limit inferior of a real-valued sequence
8.4 Monotonic sequences of real numbers
8.5 Infinite series
8.6 Inserting and removing parentheses
8.7 Alternating series
8.8 Absolute and conditional convergence
8.9 Real and imaginary parts of a complex series
8.10 Tests for convergence of series with positive terms
8.11 The geometric series
8.12 The integral test
8.13 The big oh and little oh notation
8.14 The ratio test and the root test
8.15 Dirichlet's test and Abel's test
8.16 Partial sums of the geometric series Y. z" on the unit circle Iz1 = 1
8.17 Rearrangements of series
8.18 Riemann's theorem on conditionally convergent series
8.19 Subseries
8.20 Double sequences
8.21 Double series
8.22 Rearrangement theorem for double series
8.23 A sufficient condition for equality of iterated series
8.24 Multiplication of series
8.25 Cesaro summability
8.26 Infinite products
8.27 Euler's product for the Riemann zeta function
Exercises
Chapter 9 Sequences of Functions
9.1 Pointwise convergence of sequences of functions
9.2 Examples of sequences of real-valued functions
9.3 Definition of uniform convergence
9.4 Uniform convergence and continuity
9.5 The Cauchy condition for uniform convergence
9.6 Uniform convergence of infinite series of functions
9.7 A space-filling curve
9.8 Uniform convergence and Riemann-Stieltjes integration
9.9 Nonuniformly convergent sequences that can be integrated term by term
9.10 Uniform convergence and differentiation
9.11 Sufficient conditions for uniform convergence of a series
9.12 Uniform convergence and double sequences
9.13 Mean convergence
9.14 Power series
9.15 Multiplication of power series
9.16 The substitution theorem
9.17 Reciprocal of a power series
9.18 Real power series
9.19 The Taylor's series generated by a function
9.20 Bernstein's theorem
9.21 The binomial series
9.22 Abel's limit theorem
9.23 Tauber's theorem
Exercises
Chapter 10 The Lebesgue Integral
10.1 Introduction
10.2 The integral of a step function
10.3 Monotonic sequences of step functions
10.4 Upper functions and their integrals
10.5 Riemann-integrable functions as examples of upper functions
10.6 The class of Lebesgue-integrable functions on a general interval
10.7 Basic properties of the Lebesgue integral
10.8 Lebesgue integration and sets of measure zero
10.9 The Levi monotone convergence theorems
10.10 The Lebesgue dominated convergence theorem
10.11 Applications of Lebesgue's dominated convergence theorem
10.12 Lebesgue integrals on unbounded intervals as limits of integrals on bounded intervals
10.13 Improper Riemann integrals
10.14 Measurable functions
10.15 Continuity of functions defined by Lebesgue integrals
10.16 Differentiation under the integral sign
10.17 Interchanging the order of integration
10.18 Measurable sets on the real line
10.19 The Lebesgue integral over arbitrary subsets of R
10.20 Lebesgue integrals of complex-valued functions
10.21 Inner products and norms
10.22 The set L2(I) of square-integrable functions
10.23 The set L2(I) as a semimetric space
10.24 A convergence theorem for series of functions in L2(I)
10.25 The Riesz-Fischer theorem
Exercises
Chapter 11 Fourier Series and Fourier Integrals
11.1 Introduction
11.2 Orthogonal systems of functions
11.3 The theorem on best approximation
11.4 The Fourier series of a function relative to an orthonormal system
11.5 Properties of the Fourier coefficients
11.6 The Riesz-Fischer theorem
11.7 The convergence and representation problems for trigonometric series
11.8 The Riemann-Lebesgue lemma
11.9 The Dirichlet integrals
11.10 An integral representation for the partial sums of a Fourier series
11.11 Riemann's localization theorem
11.12 Sufficient conditions for convergence of a Fourier series at a particular point
11.13 Ceshro summability of Fourier series
11.14 Consequences of Fej6r's theorem
11.15 The Weierstrass approximation theorem
11.16 Other forms of Fourier series
11.17 The Fourier integral theorem
11.18 The exponential form of the Fourier integral theorem
11.19 Integral transforms
11.20 Convolutions
11.21 The convolution theorem for Fourier transforms
11.22 The Poisson summation formula
Exercises
Chapter 12 Multivariable Differential Calculus
12.1 Introduction
12.2 The directional derivative
12.3 Directional derivatives and continuity
12.4 The total derivative
12.5 The total derivative expressed in terms of partial derivatives
12.6 An application to complex-valued functions
12.7 The matrix of a linear function
12.8 The Jacobian matrix
12.9 The chain rule
12.10 Matrix form of the chain rule
12.11 The Mean-Value Theorem for differentiable functions
12.12 A sufficient condition for differentiability
12.13 A sufficient condition for equality of mixed partial derivatives
12.14 Taylor's formula for functions from R" to RI
Exercises
Chapter 13 Implicit Functions and Extremum Problems
13.1 Introduction
13.2 Functions with nonzero Jacobian determinant
13.3 The inverse function theorem
13.4 The implicit function theorem
13.5 Extrema of real-valued functions of one variable
13.6 Extrema of real-valued functions of several variables
13.7 Extremum problems with side conditions
Exercises
Chapter 14 Multiple Riemann Integrals
14.1 Introduction
14.2 The measure of a bounded interval in R"
14.3 The Riemann integral of a bounded function defined on a compact interval in R"
14.4 Sets of measure zero and Lebesgue's criterion for existence of a multiple Riemann integral
14.5 Evaluation of a multiple integral by iterated integration
14.6 Jordan-measurable sets in R"
14.7 Multiple integration over Jordan-measurable sets
14.8 Jordan content expressed as a Riemann integral
14.9 Additive property of the Riemann integral
14.10 Mean-Value Theorem for multiple integrals
Exercises
Chapter 15 Multiple Lebesgue Integrals
15.1 Introduction
15.2 Step functions and their integrals
15.3 Upper functions and Lebesgue-integrable functions
15.4 Measurable functions and measurable sets in R'.
15.5 Fubini's reduction theorem for the double integral of a step function
15.6 Some properties of sets of measure zero
15.7 Fubini's reduction theorem for double integrals
15.8 The Tonelli-Hobson test for integrability
15.9 Coordinate transformations
15.10 The transformation formula for multiple integrals
15.11 Proof of the transformation formula for linear coordinate transformations
15.12 Proof of the transformation formula for the characteristic function of a compact cube
15.13 Completion of the proof of the transformation formula
Exercises
Chapter 16 Cauchy's Theorem and the Residue Calculus
16.1 Analytic functions
16.2 Paths and curves in the complex plane
16.3 Contour integrals
16.4 The integral along a circular path as a function of the radius
16.5 Cauchy's integral theorem for a circle
16.6 Homotopic curves
16.7 Invariance of contour integrals under homotopy
16.8 General form of Cauchy's integral theorem
16.9 Cauchy's integral formula
16.10 The winding number of a circuit with respect to a point
16.11 The unboundedness of the set of points with winding number zero
16.12 Analytic functions defined by contour integrals
16.13 Power-series expansions for analytic functions
16.14 Cauchy's inequalities. Liouville's theorem
16.15 Isolation of the zeros of an analytic function
16.16 The identity theorem for analytic functions
16.17 The maximum and minimum modulus of an analytic function
16.18 The open mapping theorem
16.19 Laurent expansions for functions analytic in an annulus
16.20 Isolated singularities
16.21 The residue of a function at an isolated singular point
16.22 The Cauchy residue theorem
16.23 Counting zeros and poles in a region
16.24 Evaluation of real-valued integrals by means of residues
16.25 Evaluation of Gauss's sum by residue calculus
16.26 Application of the residue theorem to the inversion formula for Laplace transforms
16.27 Conformal mappings
Exercises
Index of Special Symbols
Index
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