## Linear Operators: General theory |

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The

The

**restriction**of a function f to a subset A of its domain is sometimes denoted by f | A . If f : A → B , and for each bef ( A ) there is only one a € A with f ( a ) = b , f is said to have an inverse or to be one - to - one .Page 101

rather than on its domain S. Thus if there is a u - null set N such that the

rather than on its domain S. Thus if there is a u - null set N such that the

**restriction**of f to S - N is bounded , then f is said to be u - essentially bounded or simply essentially bounded . The quantity inf sup \ | ( 8 ) ] , N SES ...Page 166

Since \ in ( • ) / p converges to \ | ( ) lp in My - measure , it follows from Definition 2.17 that \ 1 ( • ) ? is un - integrable and thus that fe Lu ) . Q.E.D. In addition to the

Since \ in ( • ) / p converges to \ | ( ) lp in My - measure , it follows from Definition 2.17 that \ 1 ( • ) ? is un - integrable and thus that fe Lu ) . Q.E.D. In addition to the

**restriction**of u to a subfield of ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

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