## Linear Operators, Part 1 |

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Page 168

Let ( S , E , u ) be a positive measure space and G a

E , u , X ) , where 1 p < .0 . Then there is a set s , in E , a sub o - field of E ( S ) ,

and a closed

has ...

Let ( S , E , u ) be a positive measure space and G a

**separable**subset of L , ( S ,E , u , X ) , where 1 p < .0 . Then there is a set s , in E , a sub o - field of E ( S ) ,

and a closed

**separable**subspace X1 of 2 such that the restriction My of u to £ ,has ...

Page 501

If the compact Hausdorff space S has a countable base the space C ( S ) is

space . If g ( x , y ) is a metric in S let fn ( x ) = Q ( x , xn ) where { Xn } is a

denumerable ...

If the compact Hausdorff space S has a countable base the space C ( S ) is

**separable**. Proof . By Theorems 1.6.19 and 1.6.12 , S is a**separable**metricspace . If g ( x , y ) is a metric in S let fn ( x ) = Q ( x , xn ) where { Xn } is a

denumerable ...

Page 507

may be noted that the next theorem applies to every continuous linear map of Lj (

S , E , u ) into a

finite positive measure space , and let T be a weakly compact operator on L ( S ...

may be noted that the next theorem applies to every continuous linear map of Lj (

S , E , u ) into a

**separable**reflexive space . 10 THEOREM . Let ( S , E , u ) be a o -finite positive measure space , and let T be a weakly compact operator on L ( S ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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