## Linear Operators: General theory |

### From inside the book

Results 1-3 of 93

Page 240

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

the norm 121 = sup | Exil n i = 1 is finite . 11 . ... SES A

measurable if p - 1 ( A ) e for every Borel set A in the range of f . It is clear that ...

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

**scalars**for whichthe norm 121 = sup | Exil n i = 1 is finite . 11 . ... SES A

**scalar**function f on S is E -measurable if p - 1 ( A ) e for every Borel set A in the range of f . It is clear that ...

Page 256

... 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

( • , • ) .

... the

**scalar**product in Xi . 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( • , • ) .

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 |

Algebraic Preliminaries | 34 |

Copyright | |

21 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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