## Linear Operators: General theory |

### From inside the book

Results 1-3 of 58

Page 108

A //-

the form n / = 1 xiXEi . i=l where E( = /-1 («,□), i = l n, are disjoint sets in £ with

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

A //-

**simple function**is ju-integrable if it differs by a null function from a function ofthe form n / = 1 xiXEi . i=l where E( = /-1 («,□), i = l n, are disjoint sets in £ with

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

Page 165

Since a /^,-

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

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

measure, ...

Since a /^,-

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

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

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

Page 322

We now proceed to develop a theory of integration of scalar

respect to the vector measure p. ... A scalar valued

is ...

We now proceed to develop a theory of integration of scalar

**functions**withrespect to the vector measure p. ... A scalar valued

**function**/ defined on S is ft-**simple**if it is a finite linear combination of characteristic**functions**of sets in Z*; thisis ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

80 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

a-finite Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space valued function vector space weak topology weakly compact weakly sequentially compact zero