## Linear Operators: General theory |

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Page 240

The space bs is the linear space of all sequences x = { an } of

the norm n | x = sup ail n i = 1 n is finite . ... of characteristic functions of sets in E.

The norm in B ( S , E ) is given by the formula sup \ | ( 8 ) SES A

on ...

The space bs is the linear space of all sequences x = { an } of

**scalars**for whichthe norm n | x = sup ail n i = 1 n is finite . ... of characteristic functions of sets in E.

The norm in B ( S , E ) is given by the formula sup \ | ( 8 ) SES A

**scalar**function fon ...

Page 256

is the

always given by ( iii ) . To summarize , we state the following definition . 17

DEFINITION . For each i 1 , ... , n , let Hi be a Hilbert space with

, ) i .

is the

**scalar**product in X ;. Thus the norm in a direct sum of Hilbert spaces isalways given by ( iii ) . To summarize , we state the following definition . 17

DEFINITION . For each i 1 , ... , n , let Hi be a Hilbert space with

**scalar**products ( ·, ) i .

Page 323

A

sequence { { n } of simple functions such that ( i ) fn ( s ) converges to f ( s ) u -

almost everywhere ; ( ii ) the sequence { Sein ( s ) u ( ds ) } converges in the norm

of X ...

A

**scalar**valued measurable function f is said to be integrable if there exists asequence { { n } of simple functions such that ( i ) fn ( s ) converges to f ( s ) u -

almost everywhere ; ( ii ) the sequence { Sein ( s ) u ( ds ) } converges in the norm

of X ...

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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