## Functional AnalysisClassic exposition of modern theories of differentiation and integration and the principal problems and methods of handling integral equations and linear functionals and transformations. Topics include Lebesque and Stieltjes integrals, Hilbert and Banach spaces, self-adjunct transformations, spectral theories for linear transformations of general type, more. Translated from 2nd French edition by Leo F. Boron. 1955 edition. Bibliography. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

DIFFERENTIATION | 3 |

Some Immediate Consequences of Lebesgues Theorem | 11 |

Interval Functions | 19 |

THE LEBESGUE INTEGRAL Definition and Fundamental Properties | 29 |

The Integral for Summable Functions | 31 |

TermbyTerm Integration of an Increasing Sequence Beppo Levis Theorem | 33 |

TermbyTerm Integration of a Majorited Sequence Le besgues Theorem | 36 |

Theorems Affirming the Integrability of a Limit Function | 38 |

Uniqueness of the Generating Function | 111 |

Extension of a Linear Functional | 112 |

The Approximation Theorem Moment Problems | 115 |

Integration by Parts The Second Theorem of the Mean | 118 |

Sequences of Functionals | 119 |

Generalitation of the Stieltjes Integral | 122 |

Reduction of the LebesgueStieltjes Integral to That of Lebesgue | 124 |

Relations Between Two LebesgueStieltjes Integrals | 126 |

The Schwart Holder and Minkowski Inequalities | 40 |

The Derivative Over a Net of a Nonnegative Additive | 41 |

Measurable Sets and Measurable Functions | 43 |

Indefinite Integrals Absolutely Continuous Functions | 47 |

Example of a Monotonic Continuous Function Whose Derivative Is Zero Almost Everywhere | 48 |

Absolutely Continuous Functions Canonical Decomposition of Monotonic Functions | 50 |

Integration by Parts and Integration by Substitution | 54 |

The Integral as a Set Function | 56 |

The Space L1 and its Linear Functionals L Spaces | 57 |

Weak Convergence | 60 |

Linear Functionals | 61 |

Sequence of Linear Functionals a Theorem of Osgood | 63 |

Separability of La The Theorem of Choice | 64 |

Orthonormal Systems | 66 |

Subspaces of L The Decomposition Theorem | 71 |

Another Proof of the Theorem of Choice Extension of Functionals | 72 |

The Space L and Its Linear Functionals | 73 |

A Theorem on Mean Convergence | 78 |

A Theorem of Banach and Saks | 80 |

Functions of Several Variables | 81 |

Successive Integrations Fubinis Theorem | 83 |

Rectangle Function Parallel Displacement of the Net | 84 |

Rectangle Functions of Bounded Variation Conjugate Nets | 87 |

Additive Set Functions Sets Measurable B | 89 |

Other Definitions of the Lebesgue Integral | 92 |

Functions Measurable L and the Integral L | 94 |

Other Definitions Egoroffs Theorem | 96 |

Elementary Proof of the Theorems of Arxela and Osgood | 100 |

The Lebesgue Integral Considered as the Inverse Operation of Differentiation | 103 |

THE STIELTJES INTEGRAL AND ITS GENERALIZATIONS Linear Functionals on the Space of Continuous Functions | 105 |

Linear Functionals on the Space C | 106 |

Functions of Several Variables Direct Definition | 128 |

Definition by Means of the Principle of Transition | 130 |

The Daniel Integral | 132 |

Functionals of Variable Sign | 134 |

The Derivative of One Linear Functional With Respect to Another | 137 |

INTEGRAL EQUATIONS | 143 |

The Fredholm Alternative | 161 |

Fredholm Determinants | 172 |

Applications to Potential Theory | 190 |

Hilbert Space | 195 |

Banach Spaces | 210 |

BOUNDED SYMMETRIC UNITARY | 261 |

COMPLETELY CONTINUOUS SYMMETRIC | 227 |

Solution of the Functional Equation L4 g | 235 |

Transformations with Symmetric Kernel | 242 |

Unitary and Normal Transformations | 280 |

Unitary Transformations of the Space L1 | 291 |

SelfAdjoint Transformations Spectral Decomposition | 308 |

Extensions of Symmetric Transformations | 325 |

SELFADJOINT TRANSFORMATIONS | 341 |

Characteristic Properties of Functions of a SelfAdjoint | 351 |

The Spectrum of a SelfAdjoint Transformation and Its Pertur | 360 |

Unitary Transformations | 380 |

NonUnitary Transformations | 393 |

Ergodic Theorems | 406 |

SPECTRAL THEORIES FOR LINEAR | 415 |

Von Neumanns Theory of Spectral Sets | 435 |

Bibliography | 447 |

Appendix | 457 |

493 | |

### Common terms and phrases

absolutely continuous adjoint analogous arbitrary assume Banach space belongs bounded linear transformation bounded variation characteristic elements characteristic function characteristic value completely continuous condition consequently consider continuous function corresponding decomposition defined definition denote denumerable derivative domain element g entire space equal exists extension fact finite number finite rank follows formula function f(x functions of bounded hence Hilbert space hypothesis implies indefinite integral inequality infinite inverse kernel Lebesgue Lebesgue integral lemma limit linear combinations linear functional linear transformation ll/ll Math mation measurable function measure tero nondecreasing norm obtain obviously orthogonal orthonormal system particular permutable polynomial positive problem proved rectangle relation respect RIESZ satisfies self-adjoint transformation semi-group solution space 93 spectral family spectral set spectrum step functions Stieltjes Stieltjes integral subspace sufficient summable function symmetric transformation theorem total variation uniformly unitary transformations valid variable weak convergence xero