## Linear Operators: General theory |

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Page 168

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

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

) , and a closed

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

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

) , and a closed

**separable**subspace X , of 2 such that the restriction My of u to E ...Page 426

If X is

—y * ) xm | 8 ( 2 * , y * ) n = 1 21 1 + ( x * -- ** ) x It is easy to verify that the

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

, by ...

If X is

**separable**, let { xn } be a countable dense subset of X , and define 1 | ( x *—y * ) xm | 8 ( 2 * , y * ) n = 1 21 1 + ( x * -- ** ) x It is easy to verify that the

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

, by ...

Page 504

Theorem 6 remains valid if the hypothesis that X * is the conjugate space of a

an arbitrary B - space and T has a

will ...

Theorem 6 remains valid if the hypothesis that X * is the conjugate space of a

**separable**space is replaced by the hypothesis that X * is the conjugate space ofan arbitrary B - space and T has a

**separable**range . The proof of the corollarywill ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

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