## 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 SU

Emm , 2 ( Em , n ) Sû ( En ) + / 2 " + 1 . m = 1 m = 1 Then Ü Em , n 2 E and thus m

...

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 SU

Emm , 2 ( Em , n ) Sû ( En ) + / 2 " + 1 . m = 1 m = 1 Then Ü Em , n 2 E and thus m

...

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

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

finite subset of X , and € > 0 is

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

for every ...

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

**arbitrary**finite subset of X , and € > 0 is

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

for every ...

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### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

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