## Linear Operators: General theory |

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Page 91

admits an

invariant metric and is complete under each invariant metric . Thus every

complete linear metric space can be metrized to be an F - space . Further , a

normed linear ...

admits an

**equivalent**metric under which it is complete , then G admits aninvariant metric and is complete under each invariant metric . Thus every

complete linear metric space can be metrized to be an F - space . Further , a

normed linear ...

Page 311

PROOF . Consider the closed subspace B ( S , ) of B ( S ) . According to

Theorems 6.18 and 6.20 there is a compact Hausdorff space S , such that B ( S ,

E ) is

isomorphism x ...

PROOF . Consider the closed subspace B ( S , ) of B ( S ) . According to

Theorems 6.18 and 6.20 there is a compact Hausdorff space S , such that B ( S ,

E ) is

**equivalent**to C ( S ) . Theorem 5.1 shows that there is an isometricisomorphism x ...

Page 347

( b ) Show that L ( S , E , u ) is

collection of atoms of finite measure { En } in such that every measurable subset

of S - un En is either an atom of infinite measure or a null set . 50 Show that no ...

( b ) Show that L ( S , E , u ) is

**equivalent**to l if and only if there exists a countablecollection of atoms of finite measure { En } in such that every measurable subset

of S - un En is either an atom of infinite measure or a null set . 50 Show that no ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains 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 Hilbert space 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