## Linear Operators, Part 1 |

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Page 164

Since the extension of a

follows that { ūn ( E ) } is a bounded non - decreasing set of real numbers for each

E € £ . We define 17 ( E ) = limnûn ( E ) , E € £ . By Corollary 4 , 27 is countably ...

Since the extension of a

**non**-**negative**set function on { to E , is**non**-**negative**, itfollows that { ūn ( E ) } is a bounded non - decreasing set of real numbers for each

E € £ . We define 17 ( E ) = limnûn ( E ) , E € £ . By Corollary 4 , 27 is countably ...

Page 179

Let ( S , E , u ) be a positive measure space , f a

function defined on S and 2 ( E ) = ( t ( s ) u ( ds ) , E €E . Let g be a

a - measurable function defined on S . Then fg is u - measurable , and 58 ( s ) ?

Let ( S , E , u ) be a positive measure space , f a

**nonnegative**- measurablefunction defined on S and 2 ( E ) = ( t ( s ) u ( ds ) , E €E . Let g be a

**non**-**negative**a - measurable function defined on S . Then fg is u - measurable , and 58 ( s ) ?

Page 314

A subset K of ba ( S , 2 ) is weakly sequentially compact if and only if there exists

a

Κ . PROOF . Let KC ba ( S , E ) be weakly sequentially compact and let V = UT ...

A subset K of ba ( S , 2 ) is weakly sequentially compact if and only if there exists

a

**non**-**negative**u in ba ( S , E ) such that lim 1 ( E ) = 0 H ( E ) - > uniformly for λεΚ . PROOF . Let KC ba ( S , E ) be weakly sequentially compact and let V = UT ...

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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