## Linear Operators, Part 1 |

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Page 263

Since G is an

F1 - 6 ) . If F is a closed set it follows from this inequality , by allowing G , to range

over all open sets containing FF , that Mi ( Fi ) = Mz ( FF ; ) + Mz ( F1 - F ) .

Since G is an

**arbitrary**open set containing Fi - G , we have M ( F1 ) S2 ( G ) + M (F1 - 6 ) . If F is a closed set it follows from this inequality , by allowing G , to range

over all open sets containing FF , that Mi ( Fi ) = Mz ( FF ; ) + Mz ( F1 - F ) .

Page 269

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 M ( Fi ) 5M ( FF1 ) + 7 ( F1 - F ) . If E is an ...

Since G is an

**arbitrary**open set containing Fi - G , we have M ( F1 ) 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 FF1 , that M ( Fi ) 5M ( FF1 ) + 7 ( F1 - F ) . If E is an ...

Page 476

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

finite subset of X , and ε > 0 is

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

for every x in ...

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

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

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

for every x in ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm obtained operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero