## Linear Operators: General theory |

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Page 150

Conversely , suppose that E is a null set such that fr ( s ) → } ( s ) for s¢ E . Let

Exm = { s \ ¢ E , \ / r ( s ) - 1 ( s ) < 1 / m for r 2 ... E , u ) be a measure space and let

{ In } be a sequence of functions in

everywhere ...

Conversely , suppose that E is a null set such that fr ( s ) → } ( s ) for s¢ E . Let

Exm = { s \ ¢ E , \ / r ( s ) - 1 ( s ) < 1 / m for r 2 ... E , u ) be a measure space and let

{ In } be a sequence of functions in

**Lp**(**S**, E , p , X ) converging almosteverywhere ...

Page 524

If illa = too for all a , 0 < a si , the conclusion is trivial , so we suppose that f €

Let L , denote the class of u - integrable simple functions and ge Lo , then Hölder

...

If illa = too for all a , 0 < a si , the conclusion is trivial , so we suppose that f €

**Lp**(**S**, E , u ) for some po . By Lemma III . 8 . 5 , we may suppose that S is o - finite .Let L , denote the class of u - integrable simple functions and ge Lo , then Hölder

...

Page 663

Since the points in L ( S , E , u ) are not functions but classes of equivalent

functions , it is seen that T may not be regarded as being defined on

) unless f ( p ( s ) ) = g ( 9 ( s ) ) almost everywhere whenever f ( s ) = g ( s ) almost

...

Since the points in L ( S , E , u ) are not functions but classes of equivalent

functions , it is seen that T may not be regarded as being defined on

**LP**(**S**, E , u) unless f ( p ( s ) ) = g ( 9 ( s ) ) almost everywhere whenever f ( s ) = g ( s ) almost

...

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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 |

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