## Linear Operators: General theory |

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The following

The following

**lemma**is an obvious consequence of Definition 1 . 2**Lemma**. The intersection of an arbitrary family of convex subsets of the linear space X is convex. As examples of convex subsets of X, we note the subspaces of BE, ...Page 697

11

11

**LEMMA**. Let ( S , E , u ) be a positive measure space and let { T ( 4 , ... , tx ) , 41 , ... , tx > 0 } be a strongly measurable semi - group of operators in L ( S , E , u ) with T ( tj , ... , tx ) , 51 , \ T ( 1 , ... , telo sl .Page 699

2 Therefore 1 poe - 1 / 43 e - xt dx 2 Va Jo e - vi 2-3 / 2e - Cave - vat ) ' dx x3 / 2 evado Soon 2νπ 2e - Vt poo Va Jo 2y2 Vt - - ) dy e 2e - Vt A = e - vt , which proves ( * ) and completes the proof of the

2 Therefore 1 poe - 1 / 43 e - xt dx 2 Va Jo e - vi 2-3 / 2e - Cave - vat ) ' dx x3 / 2 evado Soon 2νπ 2e - Vt poo Va Jo 2y2 Vt - - ) dy e 2e - Vt A = e - vt , which proves ( * ) and completes the proof of the

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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