## 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 . Iff : A → B and C CA , the symbol

that is , for every a € A , the

**function**| 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 . Iff : A → B and C CA , the symbol

**f**( C ) is used for the set of all ...Page 103

space of all functions which map S into X ( see 1 . 6 . 1 ) . Unfortunately , this is

rarely the case and so a slight detour will be made . 3 DEFINITION . The

on S to X is said to be a u - null function or , when u is understood , simply a null ...

space of all functions which map S into X ( see 1 . 6 . 1 ) . Unfortunately , this is

rarely the case and so a slight detour will be made . 3 DEFINITION . The

**function f**on S to X is said to be a u - null function or , when u is understood , simply a null ...

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

members coincide n - 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 n - 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 ...

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