## Elements of the Theory of Functions and Functional Analysis, Volume 1Based on the authors' courses and lectures, this two-part advanced-level text is now available in a single volume. Topics include metric and normed spaces, continuous curves in metric spaces, measure theory, Lebesque intervals, Hilbert space, and more. Each section contains exercises. Lists of symbols, definitions, and theorems. 1957 edition. |

### What people are saying - Write a review

User Review - Flag as inappropriate

a masterpiece in math.

### Contents

The Concept of Set Operations on Sets | 1 |

Finite and Infinite Sets Denumerability | 3 |

Equivalence of Sets | 6 |

The Nondenumerability of the Set of Real Numbers | 8 |

The Concept of Cardinal Number | 9 |

Partition into Classes | 11 |

Mappings of Sets General Concept of Function | 13 |

CHAPTER II | 16 |

Linear Functionals | 77 |

The Conjugate Space | 81 |

Extension of Linear Functionals | 86 |

The Second Conjugate Space | 88 |

Weak Convergence | 90 |

Weak Convergence of Linear Functionals | 92 |

Linear Operators | 95 |

Spectrum of an Operator Resolvents | 110 |

Convergence of Sequences Limit Points | 23 |

Open and Closed Sets | 26 |

Open and Closed Sets on the Real Line | 31 |

Continuous Mappings Homeomorphism Isometry | 33 |

Complete Metric Spaces | 36 |

The Principle of Contraction Mappings and its Applications | 43 |

Applications of the Principle of Contraction Mappings in Analysis | 46 |

Compact Sets in Metric Spaces | 51 |

Arzelas Theorem and its Applications | 53 |

Compacta | 57 |

Real Functions in Metric Spaces | 62 |

Continuous Curves in Metric Spaces | 66 |

CHAPTER III | 71 |

Convex Sets in Normed Linear Spaces | 74 |

Linear Operator Equations Fredholms Theorems | 117 |

LIST OF DEFINITIONS | 123 |

Preface V | |

Collections of sets 15 | 15 |

Extension of Jordan measure 23 | 25 |

The Lebesgue extension of a measure defined on a semiring with | 31 |

CHAPTER VI | 38 |

CHAPTER VII | 48 |

Passage to the limit under the Lebesgue integral 56 | 56 |

Comparison of the Lebesgue and Riemann integrals 62 | 62 |

The representation of plane measure in terms of the linear meas | 68 |

The integral as a set function 77 | 77 |

### Common terms and phrases

additive algebra analysis applications arbitrary assume axiom Banach space belong bounded called Chapter choose clear closed interval closed sets coincide collection compact complete completely continuous concept consequently consider consisting construct contains continuous functions converges correspondence countable covering curve defined definition denote dense denumerable elementary elements equal equation equicontinuous equivalent example EXERCISES exists extension fact finite function f(x fundamental given Hence implies inequality infinite integral intersection introduce inverse least Lebesgue measure lemma limit linear functional linear operator linear space mapping means measurable sets measure metric space natural necessary norm obtain obviously operator orthogonal plane possible problems Proof properties prove real line rectangles relation respect result Riemann integrable ring sense sequence simple functions solution sphere subsequence subset subspace sufficient Suppose Theorem theory tion union values vector zero