## Linear Operators, Part 1 |

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Page 193

JS IJT 12

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

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

assumed ...

JS IJT 12

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

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

**Lemma**11 it may beassumed ...

Page 410

1 Definition. A set K Q X is convex ifx,ye K, and 0 ^ a ^ 1, imply ax-\-(l — a)y e K.

The following

intersection of an arbitrary family of convex subsets of the linear space X is

convex.

1 Definition. A set K Q X is convex ifx,ye K, and 0 ^ a ^ 1, imply ax-\-(l — a)y e K.

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.

Page 697

Nelson Dunford, Jacob T. Schwartz. 11

measure space and let { T ( 4 , . . . , tk ) , t , . . . , tx > 0 } be a strongly measurable

semi - group of operators in L ( S , E , u ) with T ( , . . . , tx ) li = 1 , \ T ( , . . . , txllo S

1 .

Nelson Dunford, Jacob T. Schwartz. 11

**LEMMA**. Let ( S , E , u ) be a positivemeasure space and let { T ( 4 , . . . , tk ) , t , . . . , tx > 0 } be a strongly measurable

semi - group of operators in L ( S , E , u ) with T ( , . . . , tx ) li = 1 , \ T ( , . . . , txllo S

1 .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm obtained operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero