## Linear Operators: General theory |

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Page 410

The following

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, and

the ...

The following

**lemma**is an obvious consequence of Definition 1 . 2**Lemma**. Theintersection 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, and

the ...

Page 697

Nelson Dunford, Jacob T. Schwartz. 11

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 .

Nelson Dunford, Jacob T. Schwartz. 11

**LEMMA**. Let ( S , E , u ) be a positivemeasure 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

Q.E.D. We shall now state and prove the

reasons occurring later the following

positive sub - semi - group rather than for a positive semi - group . The proof of ...

Q.E.D. We shall now state and prove the

**lemma**referred to as CPk . For technicalreasons occurring later the following

**lemma**is stated for what might be called apositive sub - semi - group rather than for a positive semi - group . 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