## Linear Operators, Part 1 |

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Page 169

8 Show that S can be the union of an increasing sequence of null sets even if u is

not identically

subset of X , and if f - 1 ( G ) is in for each open subset G of X , then f is totally u ...

8 Show that S can be the union of an increasing sequence of null sets even if u is

not identically

**zero**. 9 Show that if f is defined on S and has values in a compactsubset of X , and if f - 1 ( G ) is in for each open subset G of X , then f is totally u ...

Page 204

1 , Sek ! , ) 4 ( descon ) - - - } Max ( dscom ) is defined for Mo , - almost all so in Sq

. Since u ( Fn ) = Ss , In ( San ) Ha , ( dsa ) does not converge to

sin ( sq ) = 1 it follows from the dominated convergence theorem ( 6 . 16 ) that ...

1 , Sek ! , ) 4 ( descon ) - - - } Max ( dscom ) is defined for Mo , - almost all so in Sq

. Since u ( Fn ) = Ss , In ( San ) Ha , ( dsa ) does not converge to

**zero**and since 0sin ( sq ) = 1 it follows from the dominated convergence theorem ( 6 . 16 ) that ...

Page 595

Nelson Dunford, Jacob T. Schwartz. analogues of Theorem 1 for the weak and

strong topology , which we study next . 3 THEOREM . Let f , In be in F ( T ) , and

let { f ( T ) - ( T ) } converge to

fn ...

Nelson Dunford, Jacob T. Schwartz. analogues of Theorem 1 for the weak and

strong topology , which we study next . 3 THEOREM . Let f , In be in F ( T ) , and

let { f ( T ) - ( T ) } converge to

**zero**in the weak operator topology . Suppose that {fn ...

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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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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