## 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 UB2 B.

Therefore Ā UB 2 ĀU B. Conversely , Ā UB is closed ( Lemma 4 ) , so ĀU B 2 A ...

**Statement**( b ) is self - evident . Since A 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 UB2 B.

Therefore Ā UB 2 ĀU B. Conversely , Ā UB is closed ( Lemma 4 ) , so ĀU B 2 A ...

Page 415

If 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

If 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 1.4 .Page 447

( x + ayz ) .

& , -y ) +7 ( x , y ) = f ( x , 0 ) = 0 ) · 7 ( x , y ) = 0 .

**Statement**( b ) follows from the inequality a 21 ( x + 2 ( Yu + y2 ) ) = f ( x + ayı ) + k( x + ayz ) .

**Statement**( c ) is trivial .**Statement**( d ) follows from the inequality T (& , -y ) +7 ( x , y ) = f ( 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

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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