## Linear Operators: General theory |

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Page 3

The

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 . The inverse function has domain f

...

The

**restriction**of a function f to a subset A of its domain is sometimes denoted byf | 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 . The inverse function has domain f

...

Page 101

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

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 - N where N ranges

over ...

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 - N where N ranges

over ...

Page 166

Q.E.D. In addition to the

type of

...

Q.E.D. In addition to the

**restriction**of u to a subfield of Ethere is another type of**restriction**of common occurrence in integration theory . ... discussion of this othertype of

**restriction**it is not necessary to assume that u is non - negative . Suppose...

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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