## Linear Operators, Part 1 |

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Page 138

= n

proving that u { E ) = ŽM ( Em ) . Q . E . D . i = n i = n n = 1 14 THEOREM . Let u be

a ...

**Consequently**, Lil v ( u , E ; ) < oo , and v ( u , E : ) → 0 , as n + 0 . v ( u , U E ; ) = i= n

**Consequently**, i = n 7 - 1 \ , ( E ) – EM ( Ex ) ) = \ uŪE ) ) S UM , U E . ) + 0 ,proving that u { E ) = ŽM ( Em ) . Q . E . D . i = n i = n n = 1 14 THEOREM . Let u be

a ...

Page 151

Nelson Dunford, Jacob T. Schwartz. m , 1 00 m , n + 00 m , nSalt ( s ) polu , ds ) <

and lim \ n ( s ) — t ( 3 ) ] pd ( u , ds ) = 0 n - + 00 follow easily from Corollary 13 (

b ) and Theorem 3 . 6 .

Nelson Dunford, Jacob T. Schwartz. m , 1 00 m , n + 00 m , nSalt ( s ) polu , ds ) <

and lim \ n ( s ) — t ( 3 ) ] pd ( u , ds ) = 0 n - + 00 follow easily from Corollary 13 (

b ) and Theorem 3 . 6 .

**Consequently**, lim sup \ tn - Iml , Slim sup { St . \ ta ( 8 ) ...Page 309

However Eu ( F : ) SM ( A ) < oo and

above inequality shows that B ( F . ) = 0 .

integer such that fiert1M ( F ; ) < ε and let Em + 1 = F1 , . . . , Em + s = Fr , and Em

+ r + ...

However Eu ( F : ) SM ( A ) < oo and

**consequently**lim u ( Fn ) = 0 . Thus , theabove inequality shows that B ( F . ) = 0 .

**Consequently**M ( F . ) = 0 . Let r be aninteger such that fiert1M ( F ; ) < ε and let Em + 1 = F1 , . . . , Em + s = Fr , and Em

+ r + ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 other sections not shown

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Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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