## Linear Operators: General theory |

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Page 240

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

the norm ( x ) = sup / Žail is finite . 11 . The space ... SES A

is E - measurable if f - 1 ( A ) ¢ { 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 = { en } of

**scalars**for whichthe norm ( x ) = sup / Žail is finite . 11 . The space ... SES A

**scalar**function f on Sis E - measurable if f - 1 ( A ) ¢ { for every Borel set A in the range of f . It is clear

that ...

Page 256

To summarize , we state the following definition . 17 DEFINITION . For each i = 1 ,

. . . , n , let H ; be a Hilbert space with

Hilbert spaces Hi , . . . , Hn is the linear space H = H , 0 . . . Hnin which a

To summarize , we state the following definition . 17 DEFINITION . For each i = 1 ,

. . . , n , let H ; be a Hilbert space with

**scalar**products ( : , ) i . The direct sum of theHilbert spaces Hi , . . . , Hn is the linear space H = H , 0 . . . Hnin which a

**scalar**...Page 323

A

sequence { n } of simple functions such that ( i ) In ( s ) converges to f ( s ) 4 -

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

of X for ...

A

**scalar**valued measurable function f is said to be integrable if there exists asequence { n } of simple functions such that ( i ) In ( s ) converges to f ( s ) 4 -

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

of X for ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero