## Linear Operators: General theory |

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Page 177

may be

represented as the difference of its positive and negative variations ( 4.11 ) and

so we may also

measurable ...

may be

**assumed**that 2 is real valued . A real valued set function can berepresented as the difference of its positive and negative variations ( 4.11 ) and

so we may also

**assume**that 1 is positive . Let , then , { En } be a sequence ofmeasurable ...

Page 657

The problem may be formulated in abstract terms as follows : The momentary

state of a mechanical system is described by specifying a point in a " phase

space ” S. The mechanical system is

The problem may be formulated in abstract terms as follows : The momentary

state of a mechanical system is described by specifying a point in a " phase

space ” S. The mechanical system is

**assumed**to be governed by the classical ...Page 675

We may and shall

( s ) 2 1 } . Since f € L , it follows that ula ) < oo and thus f is the sum of a

summable function fXa and a bounded function fla ' . Now , since To S1 , we have

| A ( n ) ...

We may and shall

**assume**that | 2 0. Let a ' be the complement of the set a = { s \ /( s ) 2 1 } . Since f € L , it follows that ula ) < oo and thus f is the sum of a

summable function fXa and a bounded function fla ' . Now , since To S1 , we have

| A ( n ) ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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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