## Linear Operators, Part 1 |

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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 Tln ...

**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 Tln ...

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... ( iv ) If CO ( A ) is compact , then CO ( A + B ) = CO ( A ) + CO ( B ) . Proof . The

first part of

Lemma 1.4 also shows that co ( A + B ) CCO ( A ) + co ( B ) . Now , if ye Band x =

% = 1 ...

... ( 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 .Lemma 1.4 also shows that co ( A + B ) CCO ( A ) + co ( B ) . Now , if ye Band x =

% = 1 ...

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x + ayz ) .

y ) +7 ( x , y ) 1 ( x , 0 ) = 0 ) .t ( x , y ) = 0 .

**Statement**( b ) follows from the inequality a 2f ( x + 2 ( Yi + y2 ) ) 5 k ( x + ayı ) + k (x + ayz ) .

**Statement**( c ) is trivial .**Statement**( d ) follows from the inequality t ( x ,y ) +7 ( x , y ) 1 ( x , 0 ) = 0 ) .t ( 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 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

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