## Linear Operators: General theory |

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Page 72

I.11 ) is a

Given a Cauchy sequence in X / 3 , define a subsequence for which XxXx + 1 +

31 < 2 - k , k = 1 , 2 , ... , and show that a Cauch sequence in X can be chosen ...

I.11 ) is a

**B**-**space**( or an F - space with the metric \ x + 3 ) = inf ( x + z . n ( Hint :Given a Cauchy sequence in X / 3 , define a subsequence for which XxXx + 1 +

31 < 2 - k , k = 1 , 2 , ... , and show that a Cauch sequence in X can be chosen ...

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 ž 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 ž is uniquely determined up to isometric ...

Page 238

A List of Special

With the one exception of Hilbert

complex valued functions /, g defined on a specified domain S. Here, addition

and ...

A List of Special

**Spaces**We give below a list of various special**B**- and F-**spaces**.With the one exception of Hilbert

**space**, each of them will consist of real orcomplex valued functions /, g defined on a specified domain S. Here, addition

and ...

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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