## Linear Operators: General theory |

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Page 3

that is , for every a e A , the

g : B → C , then the mapping gf : A + C is defined by the equation ( gf ) ( a ) = g ( /

( a ) ) for a € A. If f : A + B and C ÇA , the symbol / ( C ) is used for the set of all ...

that is , for every a e A , the

**function f**assigns an element f ( a ) € B. If | : A + B andg : B → C , then the mapping gf : A + C is defined by the equation ( gf ) ( a ) = g ( /

( a ) ) for a € A. If f : A + B and C ÇA , the symbol / ( C ) is used for the set of all ...

Page 196

Suppose that ( S , E , u ) is a measure space and

whose values are in L ( T , ET , 2 ) , 15p < . For each s in S ,

equivalence class

everywhere ...

Suppose that ( S , E , u ) is a measure space and

**F**is a u - measurable**function**whose values are in L ( T , ET , 2 ) , 15p < . For each s in S ,

**F**( s ) is anequivalence class

**of functions**, any pair**of**whose members coincide 2 - almosteverywhere ...

Page 199

integral Ss / ( s , t ) u ( ds ) , as a

ds ) of L ( T , ET , 2 , X ) . Proof . Let E , 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 Ss / ( 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 E , 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

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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