## Linear Operators, Part 1 |

### From inside the book

Results 1-3 of 83

Page 42

As an aid for the remainder of the proof , we demonstrate the following

: Let B , C B have the properties ( a ) if x , y e By , then xy e Bi ; ( b ) if x € B1 , then

x + 0 ; then there is a homomorphism he : B → , such that h ( x ) = 1 for X e B , .

As an aid for the remainder of the proof , we demonstrate the following

**statement**: Let B , C B have the properties ( a ) if x , y e By , then xy e Bi ; ( b ) if x € B1 , then

x + 0 ; then there is a homomorphism he : B → , such that h ( x ) = 1 for X e B , .

Page 415

If X is a linear topological space , then ( ii ) co ( A ) = co ( A ) , ( iii ) co ( QA ) = Q

CO ( A ) , ( iv ) If co ( A ) is compact , then co ( A + B ) = co ( A ) + co ( B ) . PROOF .

The first part of

If X is a linear topological space , then ( ii ) co ( A ) = co ( A ) , ( iii ) co ( QA ) = Q

CO ( A ) , ( 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 .Page 447

( x + ayz ) .

, – y ) + 1 ( x , y ) 2 7 ( x , 0 ) = 0 . ( x , y ) = 0 .

**Statement**( b ) follows from the inequality 28 ( x + 5 ( 41 + y2 ) ) 5 k ( x + ayz ) + F( x + ayz ) .

**Statement**( c ) is trivial .**Statement**( d ) follows from the inequality t ( x, – y ) + 1 ( x , y ) 2 7 ( x , 0 ) = 0 . ( 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 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

### Other editions - View all

### Common terms and phrases

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