## Linear Operators, Part 1 |

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Page 3

that is , for every a € 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 CA , the symbol f ( C ) is used for the set of all ...

that is , for every a € A , the

**function f**assigns an element f ( a ) € B. If f : 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 CA , the symbol f ( C ) is used for the set of all ...

Page 196

For each s in S , F ( s ) is an equivalence class of functions , any pair of whose

members coincide 2 - almost everywhere . If for each s we select a particular

S ...

For each s in S , F ( s ) is an equivalence class of functions , any pair of whose

members coincide 2 - almost everywhere . If for each s we select a particular

**function f**( s , :) € F ( s ) , the resulting**function f**( s , t ) defined on ( R , ER , Q ) = (S ...

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 spsoo 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 ) q (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 spsoo let L , = L , ( T , ET , 2 , X ) and

define ...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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