## Linear Operators: General theory |

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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

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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