## Linear Operators: General theory |

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12

and ( T , ET , 2 ) . Let E be a o - null set in R . Then for n - almost all t , the set E ( t

) = { s ( s , t ) € E } is a u - null set . PROOF . By

...

12

**LEMMA**. Let ( R , ER , Q ) be the product of finite measure spaces ( S , E , u )and ( T , ET , 2 ) . Let E be a o - null set in R . Then for n - almost all t , the set E ( t

) = { s ( s , t ) € E } is a u - null set . PROOF . By

**Lemma**11 it may be assumed that...

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 CPx . 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 ...

Page 700

Jo This shows that if the

for any integer m 3 k . Thus , to prove the

true if k is even . If k = 1 the

shall ...

Jo This shows that if the

**lemma**is known to be true for an integer k it is also truefor any integer m 3 k . Thus , to prove the

**lemma**, it will suffice to show that it istrue if k is even . If k = 1 the

**lemma**has already been proved (**Lemma**6 ) . Weshall ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

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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 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