## Linear Operators: General theory |

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Page 177

EC En . n - 00 n00 may be

can be represented as the difference of its positive and negative variations ( 4.11

) and so we may also

EC En . n - 00 n00 may be

**assumed**that is real valued . A real valued set functioncan be represented as the difference of its positive and negative variations ( 4.11

) and so we may also

**assume**that 2 is positive . Let , then , { En } be a ...Page 178

Thus we may and shall

2 - measurable there is a sequence { n } of simple functions converging to g ( s )

for every s in F except on a set ECF with v ( 1 , E ) = 0 ( by Corollary 6.13 ( a ) ) .

Thus we may and shall

**assume**that v ( u , F ) < oo and v ( 2 , F ) < oo . Since g is2 - measurable there is a sequence { n } of simple functions converging to g ( s )

for every s in F except on a set ECF with v ( 1 , E ) = 0 ( by Corollary 6.13 ( a ) ) .

Page 675

We may and shall

Igloo for a bounded function g in Ly . Thus in proving the lemma it may be

positive .

We may and shall

**assume**that f 20. Let a ' be the complement of the set a = { s ...Igloo for a bounded function g in Ly . Thus in proving the lemma it may be

**assumed**that f is in Lj . In view of Lemma 4 it may also be**assumed**that T ispositive .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero