## Real Analysis and Foundations, Second EditionStudents preparing for courses in real analysis often encounter either very exacting theoretical treatments or books without enough rigor to stimulate an in-depth understanding of the subject. Further complicating this, the field has not changed much over the past 150 years, prompting few authors to address the lackluster or overly complex dichotomy existing among the available texts. The enormously popular first edition of Real Analysis and Foundations gave students the appropriate combination of authority, rigor, and readability that made the topic accessible while retaining the strict discourse necessary to advance their understanding. The second edition maintains this feature while further integrating new concepts built on Fourier analysis and ideas about wavelets to indicate their application to the theory of signal processing. The author also introduces relevance to the material and surpasses a purely theoretical treatment by emphasizing the applications of real analysis to concrete engineering problems in higher dimensions. Expanded and updated, this text continues to build upon the foundations of real analysis to present novel applications to ordinary and partial differential equations, elliptic boundary value problems on the disc, and multivariable analysis. These qualities, along with more figures, streamlined proofs, and revamped exercises make this an even more lively and vital text than the popular first edition. |

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### Contents

Logic and Set Theory | 1 |

EXERCISES | 34 |

EXERCISES | 67 |

Sequences | 75 |

Series of Numbers | 95 |

EXERCISES | 122 |

Basic Topology | 129 |

EXERCISES | 149 |

EXERCISES | 252 |

EXERCISES | 278 |

Applications of Analysis to Differential Equations | 285 |

EXERCISES | 301 |

Introduction to Harmonic Analysis | 307 |

EXERCISES | 336 |

Functions of Several Variables | 345 |

EXERCISES | 375 |

EXERCISES | 175 |

Differentiation of Functions | 181 |

EXERCISES | 201 |

EXERCISES | 231 |

Sequences and Series of Functions | 237 |

EXERCISES | 414 |

A Glimpse of Wavelet Theory | 421 |

445 | |

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### Common terms and phrases

assertion assume calculate called Cantor set Cauchy Chapter choose coefficients compact set complex numbers conclude continuous function continuously differentiable converge uniformly Corollary cosine countable define definition denote derivative differentiable function differential equation element equals equivalence class exercises exists fact Figure finite fj(x follows formula Fourier series function f function with domain graph hence inequality infinite integer integrable function interval 0,1 inverse Lebesgue lemma Let f lim sup limit mathematics matrix Mean Value Theorem measurable sets metric space monotone increasing natural number nonempty Notice number system one-to-one open interval open set partial sums partition polynomial positive integer power series Proof properties Proposition Prove rational numbers real analysis real line result Riemann integrable Riemann sum Riemann-Stieltjes integral satisfies sequence series converges shows solution statement subset supremum theory uncountable upper bound variables vector wavelet Weierstrass write zero