## Linear Operators: General theory |

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Page 263

Since G is an

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

range over all open sets containing FF1 , that Mz ( F ) SM ( FF ) + u2 ( F1 - F ) .

Since G is an

**arbitrary**open set containing F1 - G we have Mi ( Fi ) S2 ( G ) + M7 (F1 -- G ) . If F is a closed set it follows from this inequality , by allowing G , to

range over all open sets containing FF1 , that Mz ( F ) SM ( FF ) + u2 ( F1 - F ) .

Page 281

Let S be an

and only if it is bounded and , together with every subsequence , converges to to

quasi - uniformly on S . PROOF . The sequence { fn } in B ( S ) converges weakly

to ...

Let S be an

**arbitrary**set . A sequence { { n } in B ( S ) converges weakly to to ifand only if it is bounded and , together with every subsequence , converges to to

quasi - uniformly on S . PROOF . The sequence { fn } in B ( S ) converges weakly

to ...

Page 476

N ( T ; A , ε ) = { R | R € B ( X , Y ) , [ ( T – R ) x1 < E , X e A } where A is an

finite subset of X , and ε > O is

generalized sequence { Ta } converges to T if and only if { T _ x } converges to Tx

for ...

N ( T ; A , ε ) = { R | R € B ( X , Y ) , [ ( T – R ) x1 < E , X e A } where A is an

**arbitrary**finite subset of X , and ε > O is

**arbitrary**. Thus , in the strong topology , ageneralized sequence { Ta } converges to T if and only if { T _ x } converges to Tx

for ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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