## Linear Operators: General theory |

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Page 103

space of all functions which map S into X ( see 1.6.1 ) . ... we define F ( p / 9 ) = 1/

9 , and set f ( x ) = 0 , SES - R . Since for each a > 0 , s ( li > ) is a finite set , f is a u

- null function . ... A

space of all functions which map S into X ( see 1.6.1 ) . ... we define F ( p / 9 ) = 1/

9 , and set f ( x ) = 0 , SES - R . Since for each a > 0 , s ( li > ) is a finite set , f is a u

- null function . ... A

**function f**on S to X is a u - null function if and only if t = 0 .Page 196

Section VI.8 ) one is faced with the following situation . Suppose that ( S , E , u ) is

a measure space and

ET , 2 ) , 13p < 0. For each s in S ,

...

Section VI.8 ) one is faced with the following situation . Suppose that ( S , E , u ) is

a measure space and

**F**is a u - measurable**function**whose values are in L ( T ,ET , 2 ) , 13p < 0. For each s in S ,

**F**( s ) is an equivalence class**of functions**, any...

Page 199

integral Sst ( s , t ) u ( ds ) , as a

ds ) of L ( T , ET , 2 , X ) . Proof . Let Eq be partitioned into a sequence { En } of

disjoint sets of finite 2 - measure . For 1 < p so let L , L , ( T , ET , 2 , X ) and define

...

integral Sst ( s , t ) u ( ds ) , as a

**function of**t , is equal to the element Ss F ( s ) u (ds ) of L ( T , ET , 2 , X ) . Proof . Let Eq be partitioned into a sequence { En } of

disjoint sets of finite 2 - measure . For 1 < p so let L , L , ( T , ET , 2 , X ) and define

...

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### Contents

Preliminary Concepts 1 1 Preliminary Concepts | 1 |

B Topological Preliminaries 10 B Topological Preliminaries | 10 |

Metric Spaces | 18 |

Copyright | |

80 other sections not shown

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