## 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 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 T / n

( . ) ...

**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 T / n

( . ) ...

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 .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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