## Linear Operators, Part 1 |

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Page 183

it

Conversely , since $ ( EU EZ ) = $ ( E U $ ( EZ ) and $ ( E E2 ) = 4 ( E ) * ( E2 ) if

Ey , E , belong to £1 , v ( uz , ° ( E ) ) is a nonnegative additive set function defined

...

it

**follows**readily from the definition of v ( ug ) that v ( uz , 6 - 1 ( E ) ) 2 ulla , E ) .Conversely , since $ ( EU EZ ) = $ ( E U $ ( EZ ) and $ ( E E2 ) = 4 ( E ) * ( E2 ) if

Ey , E , belong to £1 , v ( uz , ° ( E ) ) is a nonnegative additive set function defined

...

Page 403

It

measure defined intrinsically in any n - dimensional real Hilbert space H , without

reference to any particular coordinate system in that space . This measure will be

...

It

**follows**from the rotational invariance of Min that un may be regarded as ameasure defined intrinsically in any n - dimensional real Hilbert space H , without

reference to any particular coordinate system in that space . This measure will be

...

Page 576

It

verify that it is an isomorphism , it will suffice to show that E ( 0 ) = 0 only when o

is the void set $ . Now if E ( 0 ) = 0 , then X , = { 0 } and o ( T . ) = $ . It

...

It

**follows**from Theorem 10 that the map o → E ( 0 ) is a homeomorphism . Toverify that it is an isomorphism , it will suffice to show that E ( 0 ) = 0 only when o

is the void set $ . Now if E ( 0 ) = 0 , then X , = { 0 } and o ( T . ) = $ . It

**follows**from...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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 |

### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint 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 obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero