## Linear Operators: General theory |

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Page 106

The functions totally u - measurable on S , or , if u is understood , totally

measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u -

characteristic ...

The functions totally u - measurable on S , or , if u is understood , totally

measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u -

**simple functions**. If for every E in { with v ( u , E ) < oo , the product xet off with thecharacteristic ...

Page 108

Nelson Dunford, Jacob T. Schwartz. n 13 DEFINITION . A u -

integrable if it differs by a null function from a function of the form = I xilles , where

E ; = 7 - 1 ( xi ) , i = 1 , . . . , n , are disjoint sets in E with union S and where Xi ...

Nelson Dunford, Jacob T. Schwartz. n 13 DEFINITION . A u -

**simple function**is u -integrable if it differs by a null function from a function of the form = I xilles , where

E ; = 7 - 1 ( xi ) , i = 1 , . . . , n , are disjoint sets in E with union S and where Xi ...

Page 322

We now proceed to develop a theory of integration of scalar

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

this is ...

We now proceed to develop a theory of integration of scalar

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

**function**f defined on S is u -**simple**if it is a finite linear combination of characteristic**functions**of sets in £ * ;this is ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 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

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero