## Linear Operators: General theory |

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Page 72

11 ) is a

Hint : Given a Cauchy sequence in X / 3 , define a subsequence for which XX - Xk

+ 1 + 81 < 2 - k , k = 1 , 2 , . . . , and show that a Cauchy sequence in X can be ...

11 ) is a

**B**-**space**( or an F - space ) with the metric 18 + 3 ) = inf ( x + 21 . 2€3 (Hint : Given a Cauchy sequence in X / 3 , define a subsequence for which XX - Xk

+ 1 + 81 < 2 - k , k = 1 , 2 , . . . , and show that a Cauchy sequence in X can be ...

Page 89

In the definitions of F - and

their metric topology . ... Then X is isomorphic and isometric with a dense linear

subspace of an F - space Ž . The space X is uniquely determined up to isometric

...

In the definitions of F - and

**B**-**spaces**, we required the spaces to be complete intheir metric topology . ... Then X is isomorphic and isometric with a dense linear

subspace of an F - space Ž . The space X is uniquely determined up to isometric

...

Page 838

Ill special

Banach limits, existence and properties, II.4.22-23 (73) Banach theorem, on

convergence of measurable functions, IV. 11.2-3 (332-334) Banach-Stone

theorem, ...

Ill special

**B**-**spaces**, Chap. IV properties, IV.15 Baire category theorem, 1.6.9 (20)Banach limits, existence and properties, II.4.22-23 (73) Banach theorem, on

convergence of measurable functions, IV. 11.2-3 (332-334) Banach-Stone

theorem, ...

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad 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 meaning measure space metric neighborhood norm operator positive measure problem Proc PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero