## Linear Operators, Part 1 |

### From inside the book

Results 1-3 of 83

Page 114

integrable simple functions ; to prove it in the general case , let In be a sequence

of finitely valued u - simple functions which determine g . Then the functions Tln ...

**Statement**( b ) is clear from the definitions .**Statement**( c ) is evident for u -integrable simple functions ; to prove it in the general case , let In be a sequence

of finitely valued u - simple functions which determine g . Then the functions Tln ...

Page 415

... ( iv ) If CO ( A ) is compact , then CO ( A + B ) = CO ( A ) + CO ( B ) . Proof . The

first part of

Lemma 1.4 also shows that co ( A + B ) CCO ( A ) + co ( B ) . Now , if ye Band x =

% = 1 ...

... ( iv ) If CO ( A ) is compact , then CO ( A + B ) = CO ( A ) + CO ( B ) . Proof . The

first part of

**statement**( i ) follows in an elementary fashion from Lemma 1.4 .Lemma 1.4 also shows that co ( A + B ) CCO ( A ) + co ( B ) . Now , if ye Band x =

% = 1 ...

Page 447

x + ayz ) .

y ) +7 ( x , y ) 1 ( x , 0 ) = 0 ) .t ( x , y ) = 0 .

**Statement**( b ) follows from the inequality a 2f ( x + 2 ( Yi + y2 ) ) 5 k ( x + ayı ) + k (x + ayz ) .

**Statement**( c ) is trivial .**Statement**( d ) follows from the inequality t ( x ,y ) +7 ( x , y ) 1 ( x , 0 ) = 0 ) .t ( x , y ) = 0 .

**Statement**( e ) is trivial . Q.E.D. 4 ...### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

### Common terms and phrases

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