## Linear Operators: General theory |

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Page 240

The space bs is the linear space of all sequences x =

Am } of n i = 1 | x | = sup oil is finite . ... the formula = sup | ( 8 ) SES A

function f on S is E - measurable if f - 1 ( A ) € E for every Borel set A in the range

of f .

The space bs is the linear space of all sequences x =

**scalars**for which the norm {Am } of n i = 1 | x | = sup oil is finite . ... the formula = sup | ( 8 ) SES A

**scalar**function f on S is E - measurable if f - 1 ( A ) € E for every Borel set A in the range

of f .

Page 256

If , however , each of the spaces X1 , ... , Xn are Hilbert spaces then it will always

be understood , sometimes without explicit mention , that X is the uniquely

determined Hilbert space with

= ( Xi ...

If , however , each of the spaces X1 , ... , Xn are Hilbert spaces then it will always

be understood , sometimes without explicit mention , that X is the uniquely

determined Hilbert space with

**scalar**product n ( iv ) ( [ X1 , ... , Xn ] , 91 , ... , Yn ] )= ( Xi ...

Page 323

A

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

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

of X ...

A

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

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

of X ...

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad 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 meaning measure space metric neighborhood norm operator positive measure problem Proc PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero