## Foundations of Mathematical AnalysisThis classroom-tested volume offers a definitive look at modern analysis, with views of applications to statistics, numerical analysis, Fourier series, differential equations, mathematical analysis, and functional analysis. A self-contained text, it presents the necessary background on the limit concept. (The first seven chapters could constitute a one-semester course on introduction to limits.) Subsequent chapters discuss differential calculus of the real line, the Riemann-Stieltjes integral, sequences and series of functions, transcendental functions, inner product spaces and Fourier series, normed linear spaces and the Riesz representation theorem, and the Lebesgue integral. More than 750 exercises help reinforce the material. 1981 edition. 34 figures. |

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#### Review: Foundations of Mathematical Analysis

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### Contents

Sets and Functions | 1 |

The Real Number System | 9 |

Set Equivalence | 26 |

Sequences of Real Numbers | 34 |

Real Exponents | 55 |

The BolzanoWei erst rass Theorem | 58 |

The Cauchy Condition | 59 |

The lim sup and lirn inf of Bounded Sequences | 61 |

Closed Sets | 128 |

Open Sets | 132 |

Continuous Functions on Metric Spaces | 136 |

The Relative Metric | 141 |

Compact Metric Spaces | 144 |

The BoizanoWeierstrass Characterization of a Compact Metric Space | 148 |

Continuous Functions on Compact Metric Spaces | 152 |

Connected Metric Spaces | 155 |

The lim sup and lim inf of Unbounded Sequences | 69 |

Infinite Series | 73 |

Algebraic Operations on Series | 76 |

Series with Nonnegative Terms | 77 |

The Alternating Series Test | 80 |

Absolute Convergence | 81 |

Power Series | 87 |

Conditional Convergence | 90 |

Double Series and Applications | 92 |

Limits of RealValued Functions and Continuous Functions on the Real Line | 102 |

Limit Theorems for Functions | 105 |

OneSided and Infinite Limits | 107 |

Continuity | 109 |

The HeineBorel Theorem and a Consequence for Continuous Functions | 112 |

VH Metric Spaces | 116 |

R I2 and the CauchySchwarz Inequality | 120 |

Sequences in Metric Spaces | 125 |

Complete Metric Spaces | 159 |

Baire Category Theorem | 166 |

Differential Calculus of the Real Line | 171 |

The RiemannStieltjes Integral | 189 |

Sequences and Series of Functions | 245 |

Transcendental Functions | 268 |

Inner Product Spaces and Fourier Series | 280 |

Normed Linear Spaces and the Riesz Representation | 335 |

The Dual Space of a Normed Linear Space | 343 |

Proof of the Riesz Representation Theorem | 349 |

The Lebesgue Integral | 355 |

Vector Spaces | 405 |

Hints to Selected Exercises | 411 |

421 | |

429 | |

### Common terms and phrases

a e BV[a a-algebra axiom bounded function bounded sequence Cauchy sequence closed interval closed subset compact metric space continuous function convergent subsequence converges absolutely converges pointwise converges uniformly Corollary countable CP[a define Definition differentiable diverges equation Exercise exists a positive finite Fourier series Give an example hence If/is improper integral increasing function inequality infinite series inner product space Jx Jx least upper bound Lemma let x e Let/be lim inf lim sup limn linear transformation measurable functions measure zero nonnegative normed linear space open ball open interval open set open subset orthonormal set partial sums positive integer power series proof of Theorem rational number real numbers real sequence Riemann integral Riemann-Stieltjes integral Riesz representation theorem Section sequence of partial space and let Suppose that/is then/is theorem Theorem vector space verify