## Linear Operators: General theory |

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Page 193

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

technical reasons occurring later the following

called a 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 . Fortechnical reasons occurring later the following

**lemma**is stated for what might becalled a positive 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

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero