## Linear Operators: General theory |

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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 CCA , the symbol / ( 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 CCA , the symbol / ( C ) is used for the set of all ...

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...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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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 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