## 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

20 there is a compact Hausdorff space S , such that B ( S , 2 ) is

S ) . Theorem 5 . 1 shows that there is an isometric isomorphism æ * t u between

B * ( S , E ) and ba ( S , E ) , which is determined by the equation w * XE = u ( E ) ...

20 there is a compact Hausdorff space S , such that B ( S , 2 ) is

**equivalent**to C (S ) . Theorem 5 . 1 shows that there is an isometric isomorphism æ * t u between

B * ( S , E ) and ba ( S , E ) , which is determined by the equation w * XE = u ( E ) ...

Page 347

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

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

measurable subset of S - u , 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 4 if and only if there exists acountable collection of atoms of finite measure { En } in such that every

measurable subset of S - u , 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 |

Algebraic Preliminaries | 34 |

Copyright | |

31 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 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