## Linear Operators: General theory |

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Page 289

that there is a functional y * € L * such that *** ( ** ) = y * ( g ) when g and x * are

connected , as in Theorem 1 , by the formula « * = [ st ( s ) g ( s ) u ( ds ) , fe Lp ...

**Proof**. Let æ ** € ( L * ) * . By Theorem 1 , L * is isometrically isomorphic to Lq , sothat there is a functional y * € L * such that *** ( ** ) = y * ( g ) when g and x * are

connected , as in Theorem 1 , by the formula « * = [ st ( s ) g ( s ) u ( ds ) , fe Lp ...

Page 415

Hence ko e p - A , and thus pe A + k , C A + K . Q.E.D. Since the commutativity of

the group G is not essential to the

topological groups . LEMMA . For arbitrary sets A , B in a linear space X : ( i ) co ...

Hence ko e p - A , and thus pe A + k , C A + K . Q.E.D. Since the commutativity of

the group G is not essential to the

**proof**, the same result holds for non - Abeliantopological groups . LEMMA . For arbitrary sets A , B in a linear space X : ( i ) co ...

Page 699

The

11 as outlined in the diagram : C = CPx = DPk = Dx = Cx . The

implication CPx = DPK which is the

of ...

The

**proof**of this lemma is the most involved of all the steps in the**proof**of Lemma11 as outlined in the diagram : C = CPx = DPk = Dx = Cx . The

**proof**of theimplication CPx = DPK which is the

**proof**of Lemma 14 is very similar to the**proof**of ...

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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