## Linear Operators: General theory |

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Page 177

Nelson Dunford, Jacob T. Schwartz, William G. Bade. may be

real valued . A real valued set function can be represented as the difference of its

positive and negative variations ( 4 . 11 ) and so we may also

Nelson Dunford, Jacob T. Schwartz, William G. Bade. may be

**assumed**that 2 isreal valued . A real valued set function can be represented as the difference of its

positive and negative variations ( 4 . 11 ) and so we may also

**assume**that a is ...Page 278

In the complex case it is also

S such that * * = f ( s ) , fe C ( S ) . Proof . By the preceding lemma x * is a point in

the space S , defined in the proof of Theorem 18 . By Theorem 22 , S is ...

In the complex case it is also

**assumed**that x * 4 = x * f . Then there is a point s inS such that * * = f ( s ) , fe C ( S ) . Proof . By the preceding lemma x * is a point in

the space S , defined in the proof of Theorem 18 . By Theorem 22 , S is ...

Page 326

Evidently we may

țr was

sequence { gx } converges u - almost everywhere to f . It remains to show that the

...

Evidently we may

**assume**that Ox 52 - k . Let nx > 0 be such that if A € E ... Sincețr was

**assumed**to converge u - almost everywhere to f , we conclude that thesequence { gx } converges u - almost everywhere to f . It remains to show that the

...

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### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements 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 operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group 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