## Linear Operators, Part 1 |

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Page 11

Ā . The set A UB contains A and is closed . Thus AUBĀ , and similarly AUB2 B .

Therefore AUB2 AU B . Conversely , Āu B is closed ( Lemma 4 ) , so ĀUB 2 A UB

...

**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 AUBĀ , and similarly AUB2 B .

Therefore AUB2 AU B . Conversely , Āu B is closed ( Lemma 4 ) , so ĀUB 2 A UB

...

Page 415

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

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

. The first part of

11 X is a linear topological space , then ( ii ) co ( A ) = co ( A ) , ( iii ) co ( AA ) = 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 ) .

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

**Statement**( b ) follows from the inequality 2f ( x + ( 91 + y2 ) ) = f ( x + ayz ) + ( x +ayz ) .

**Statement**( c ) is trivial .**Statement**( d ) follows from the inequality t ( X , - y) + 1 ( x , y ) 2 ( x , 0 ) = 0 ) • 7 ( 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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Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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