## Foundations of Modern AnalysisMeasure and integration, metric spaces, the elements of functional analysis in Banach spaces, and spectral theory in Hilbert spaces — all in a single study. Only book of its kind. Unusual topics, detailed analyses. Problems. Excellent for first-year graduate students, almost any course on modern analysis. Preface. Bibliography. Index. |

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### Contents

Preface | 1 |

1 | 8 |

1 | 17 |

INTEGRATION | 29 |

METRIC SPACES | 90 |

ELEMENTS OF FUNCTIONAL ANALYSIS | 123 |

COMPLETELY CONTINUOUS OPERATORS | 186 |

HILBERT SPACES AND SPECTRAL THEORY | 201 |

4 | 213 |

244 | |

### Common terms and phrases

algebra apply assertion assume ball Banach space belong bounded bounded linear called Cauchy sequence clearly closed linear closed set compact complete conclude Consider consisting constant contains continuous function convergent Corollary countable covering defined Definition Denote dense DIFFERENTIAL disjoint domain easily eigenvalue elements equation Example exists extended finite follows function f given gives Hence Hilbert space holds implies inequality integrable function interval INTRODUCTION Lemma limit linear operator linear subspace mathematics measurable function measurable set measure space metric space monotone normed linear space Note open set outer measure positive integer Problem projection Proof properties Prove real-valued relation respect result satisfies separable sequence Similarly simple functions solution space and let subsequence subset Suppose Theorem theory uniformly union unique weakly write zero