## Linear Operators: General theory |

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Ā . The set A UB contains A and is closed . Thus À UB2 Ā , and similarly A UB ] B .

Therefore A UB ? AU B . Conversely , Ā UB is closed ( Lemma 4 ) , so ĀUB2 ...

**Statement**( b ) is self - evident . Since Ā is closed , ( Lemma 4 ) , it follows that Ā =Ā . The set A UB contains A and is closed . Thus À UB2 Ā , and similarly A UB ] B .

Therefore A UB ? AU B . Conversely , Ā UB is closed ( Lemma 4 ) , so ĀUB2 ...

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 .Page 447

+ ayz ) .

– y ) + ( x , y ) 2 t ( x , 0 ) = 0 · ( x , y ) = 0 .

**Statement**( b ) follows from the inequality a 2k ( x + 5 ( 4 + y2 ) ) = f ( x + ay . ) + ( x+ ayz ) .

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

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

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