## Integral and Functional AnalysisThis book is based on two closely-related courses. The first of these courses is Integration and Metric Spaces, and the second being Functional Analysis. Though the contents of Functional Analysis have been used for both an undergraduate course and an introductory graduate course, this text is designed primarily for undergraduate students. The prerequisites of this book are deliberately modest, and it is assumed that the students have some familiarity with Introductory Calculus and Linear Algebra plus the basic (direct, indirect) proof methods. |

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### Contents

1 | |

8 | |

15 | |

17 | |

22 Algebraic Operations and the Darboux Criterion | 23 |

23 Fundamental Theorem of Calculus | 28 |

24 Improper Integrals | 35 |

Problems | 42 |

62 Continuous Mappings on Compact or Connected Spaces | 133 |

63 Sequences of Mappings | 135 |

64 Contractions | 139 |

65 Equivalence of Metric Spaces | 144 |

Problems | 146 |

Normed Linear Spaces | 149 |

72 Finite Dimensional Spaces | 156 |

73 Bounded Linear Operators | 160 |

RiemannStieltjes Integrals | 45 |

32 Definition and Basic Properties | 48 |

33 Nonexistence and Existence for Integrals | 51 |

34 Evaluation of Integrals | 59 |

35 Improper Cases | 62 |

Problems | 66 |

LebesgueRadonStieltjes Integrals | 69 |

41 Foundational Material | 70 |

42 Essential Properties | 76 |

43 Convergence Theorems | 80 |

44 Extension via Measurability | 86 |

45 Double and Iterated Integrals with Applications | 91 |

Problems | 102 |

Metric Spaces | 109 |

52 Completeness | 115 |

53 Compactness Density and Separability | 120 |

Problems | 127 |

Continuous Maps | 129 |

74 Linear Functionals via HahnBanach Extension | 163 |

Problems | 169 |

Banach Spaces via Operators and Functionals | 173 |

82 Uniform Boundedness Open Mapping and Closed Graph | 178 |

83 Dual Banach Spaces by Examples | 182 |

84 Weak and Weakstar Topologies | 192 |

85 Compact and Dual Operators | 197 |

Problems | 200 |

Hilbert Spaces and Their Operators | 205 |

92 Orthogonality Orthogonal Complement and Duality | 208 |

93 Orthonormal Sets and Bases | 211 |

94 Five Special Bounded Operators | 217 |

95 Compact Operators via the Spectrum | 225 |

Problems | 233 |

Hints and Solutions | 239 |

281 | |

285 | |

### Common terms and phrases

Accordingly assume Banach space bijective Br(x Cauchy sequence Cauchy-Schwarz inequality choose cluster point compact operator Consequently contains continuous functions continuous mappings contraction map contradiction convergence theorem Corollary countable defined Definition denoted disjoint eigenvalues element equivalent Example exists f is continuous fc=i fc=l finite number fixed point function f given hence Hilbert space implies improper integral infinite inner product Lebesgue-Radon-Stieltjes integrals Lemma Let f Let g lim^oo linear functional linear operator linear subspace LRSg(R metric space mg-measurable mg-null nonempty nonnegative nonzero normed linear space Note open ball open sets orthonormal basis orthonormal set partition Proof Prove the following real numbers Remark Riemann integrable Riemann-Stieltjes integrals RSg[a self-adjoint space over F step functions subintervals topological triangle inequality uniformly continuous unique vectors verify yields