## Linear Operators: General theory |

### From inside the book

Results 1-3 of 58

Page 108

A ^-

the form n / = 2 XiXEi ' where Et = f~l{xt), i = 1, . . ., n, are disjoint sets in X with

union S and where x( = 0 if v(fi, E{) — oo. The phrases "^-integrable //-simple ...

A ^-

**simple function**is fi-integrable if it differs by a null function from a function ofthe form n / = 2 XiXEi ' where Et = f~l{xt), i = 1, . . ., n, are disjoint sets in X with

union S and where x( = 0 if v(fi, E{) — oo. The phrases "^-integrable //-simple ...

Page 165

Since a /^-

immeasurable function is ^-measurable. ... Ev Let / be a /^-integrable function and

let {/„} be a sequence of jUj-integrable

measure, ...

Since a /^-

**simple function**is clearly ^-simple, it follows immediately that a /immeasurable function is ^-measurable. ... Ev Let / be a /^-integrable function and

let {/„} be a sequence of jUj-integrable

**simple functions**converging to / in ^-measure, ...

Page 322

A scalar

e Z*; by III. 6. 9, this is the case if / is A-measurable. A scalar valued

defined on S is fx-

A scalar

**function**/ defined on S is fi-measurable if for every Borel set B of scalars,e Z*; by III. 6. 9, this is the case if / is A-measurable. A scalar valued

**function**/defined on S is fx-

**simple**if it is a finite linear combination of characteristic**functions**...### What people are saying - Write a review

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

### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

78 other sections not shown

### Other editions - View all

### Common terms and phrases

additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations disjoint Doklady Akad Duke Math element ergodic theorem exists finite dimensional function defined Hausdorff space Hence Hilbert space homeomorphism ibid inequality integral Lebesgue Lebesgue measure Lemma linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc Proof properties proved real numbers reflexive Riesz Russian Sbornik N. S. scalar self-adjoint semi-group sequentially compact Show simple functions spectral Studia Math subset subspace Suppose theory topological space Trans uniformly Univ valued function Vber vector space weak topology weakly compact zero