## 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

x , y ) = T ( x , 0 ) = 0 • t ( x , y ) = 0 .

DEFINITION .

**Statement**( b ) follows from the inequality - y2 ) ) = f ( x + ayı ) + k ( x + ayz ) .**Statement**( c ) is trivial .**Statement**( d ) follows from the inequality T ( x , - y ) + 1 (x , y ) = T ( x , 0 ) = 0 • t ( x , y ) = 0 .

**Statement**( e ) is trivial . Q . E . D . 4DEFINITION .

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### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

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