## Linear Operators, Part 1 |

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Page 263

Since G is an

( F1 - G7 ) . If F is a closed set it follows from this inequality , by allowing G , to

range over all open sets containing FF , that # 1 ( F ) < ! ( FF1 ) + uz ( F1 - F ) .

Since G is an

**arbitrary**open set containing F1 - G , we have My ( F1 ) S2 ( G ) +41( F1 - G7 ) . If F is a closed set it follows from this inequality , by allowing G , to

range over all open sets containing FF , that # 1 ( F ) < ! ( FF1 ) + uz ( F1 - F ) .

Page 432

To see this , let m be an

, there is an element zm € A such that 2 * ( cm ) - ** ( * ) < 1m , i 1 , ... , n . Since A

is weakly sequentially compact , a subsequence of { zm } will converge weakly ...

To see this , let m be an

**arbitrary**integer ; since x ** is in the X * -closure of x ( A ), there is an element zm € A such that 2 * ( cm ) - ** ( * ) < 1m , i 1 , ... , n . Since A

is weakly sequentially compact , a subsequence of { zm } will converge weakly ...

Page 476

... A } where A is an

strong topology , a generalized sequence { Tx } converges to T if and only if { Tqx

} converges to Tx for every x in X. 3 DEFINITION . The weak operator topology ...

... A } where A is an

**arbitrary**finite subset of X , and e > 0 is**arbitrary**. Thus , in thestrong topology , a generalized sequence { Tx } converges to T if and only if { Tqx

} converges to Tx for every x in X. 3 DEFINITION . The weak operator topology ...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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