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. |
Contents
The Concept of Set Operations on Sets | 1 |
Finite and Infinite Sets Denumerability | 5 |
Equivalence of Sets | 7 |
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 CHAPTER II | 13 |
Definition and Examples of Metric Spaces | 17 |
The Conjugate Space | 85 |
Extension of Linear Functionals | 87 |
The Second Conjugate Space | 89 |
Weak Convergence | 91 |
Weak Convergence of Linear Functionals | 93 |
Linear Operators | 97 |
Spectrum of an Operator Resolvents | 110 |
Linear Operator Equations Fredholms Theorems | 117 |
Convergence of Sequences Limit Points | 23 |
Open and Closed Sets | 27 |
Open and Closed Sets on the Real Line | 31 |
Continuous Mappings Homeomorphism Isometry | 35 |
Complete Metric Spaces | 37 |
The Principle of Contraction Mappings and its Applications 16 26 31 36 | 43 |
Applications of the Principle of Contraction Mappings in Analysis | 46 |
Compact Sets in Metric Spaces | 51 |
Compacta | 59 |
Arzelas Theorem and its Applications 19 Real Functions in Metric Spaces | 63 |
CHAPTER III | 65 |
Continuous Curves in Metric Spaces 51 | 67 |
Definition and Examples of Normed Linear Spaces | 73 |
Convex Sets in Normed Linear Spaces | 75 |
Linear Functionals | 79 |
77 | 123 |
86 | 128 |
90 | 129 |
Preface | |
Extension of Jordan measure | 9 |
Collections of sets | 17 |
Complete additivity The general problem of the extension | 28 |
CHAPTER VI | 38 |
CHAPTER VII | 48 |
Passage to the limit under the Lebesgue integral | 56 |
Comparison of the Lebesgue and Riemann integrals | 62 |
The representation of plane measure in terms of the linear meas | 68 |
The integral as a set function | 77 |
92 | 92 |
Common terms and phrases
A₁ A₂ arbitrary assume B₁ B₂ Banach space Borel algebra bounded cardinal number closed interval closed sets collection of sets compact compactum completely continuous operator consequently consider contains continuous functions contraction mappings converges a.e. converges to f(x COROLLARY corresponding definition denote denumerable E₁ elementary sets elements equation equivalent everywhere dense example exists fact finite number fn(x following theorem function ƒ functions defined Hence Hilbert space implies inequality infinite-dimensional intersection Jordan measurable L₂ Lebesgue integral lemma linear functional linear operator mapping measurable functions measurable sets metric space normed linear space o-additive obtain open set orthogonal orthonormal set p(xn plane Proof properties proves the theorem R(Sm real line real numbers rectangles Riemann integrable ring satisfy the condition semi-ring sphere subsets subspace Suppose theory vectors weak convergence zero µ(An