## Linear Operators, Part 1 |

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Page 168

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

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

a closed

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

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

a closed

**separable**subspace X of X such that the restriction My of u to ; has the ...Page 426

If X is

y * ) xn | 0 ( 2 * , 4 * ) = Σ n = 1 21 1+ | ( 2 * —Y * ) xml It is easy to verify that the

topology of S * defined by this metric is weaker than the X topology of S * . Hence

...

If X is

**separable**, let { xn } be a countable dense subset of X , and define | ( x * —y * ) xn | 0 ( 2 * , 4 * ) = Σ n = 1 21 1+ | ( 2 * —Y * ) xml It is easy to verify that the

topology of S * defined by this metric is weaker than the X topology of S * . Hence

...

Page 507

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

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 L (

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 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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