## Linear Operators: General theory |

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Page 135

Let E be the union of an

each n = 1 , 2 , . . . let the sequence { Em . n } have the properties Em , ne£ , E ,

SÜE EMC Emn ) SÃ ( E . ) + ] 2 + 1 m = 1 m = 1 Then u Em . n ? E and thus m , n =

1 ...

Let E be the union of an

**arbitrary**sequence { En } of sets in S . Let E > 0 and foreach n = 1 , 2 , . . . let the sequence { Em . n } have the properties Em , ne£ , E ,

SÜE EMC Emn ) SÃ ( E . ) + ] 2 + 1 m = 1 m = 1 Then u Em . n ? E and thus m , n =

1 ...

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 FFı , that 47 ( F1 ) " ( FF1 ) + uz ( F1 - F ) . If E is an ...

Since G is an

**arbitrary**open set containing Fi - G , we have U1 ( Fi ) S2 ( G ) + M (F1 - G ) . If F is a closed set it follows from this inequality , by allowing G , to range

over all open sets containing FFı , that 47 ( F1 ) " ( FF1 ) + uz ( F1 - F ) . If E is an ...

Page 476

... A } where A is an

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

x } 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 ε > 0 is**arbitrary**. Thus , in thestrong topology , a generalized sequence { Tx } converges to T if and only if { T «

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

...

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### Contents

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

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