## Linear Operators, Part 1 |

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Page 72

11 ) is a

Hint : Given a Cauchy sequence in X / 3 , define a subsequence for which Xx —

Xx + 1 + 3 ) < 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 1x + 31 = inf ( x + zl . 283 (Hint : Given a Cauchy sequence in X / 3 , define a subsequence for which Xx —

Xx + 1 + 3 ) < 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 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 398

Let S be a Stone space and X = C ( S ) the real continuous functions .

Grothendieck [ 4 ; p . 168 ] showed that every X - convergent sequence in X * is

actually X * * - convergent , and that if Y is a separable

operator in B ( X ...

Let S be a Stone space and X = C ( S ) the real continuous functions .

Grothendieck [ 4 ; p . 168 ] showed that every X - convergent sequence in X * is

actually X * * - convergent , and that if Y is a separable

**B**-**space**, then anyoperator in B ( X ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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### Common terms and phrases

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 obtained 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