## Linear Operators, Part 1 |

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Page 20

If every Cauchy sequence is convergent , a metric space is said to be

The next three lemmas are immediate consequences of the definitions . 6

LEMMA . In a metric space , a convergent sequence is a Cauchy sequence . A

Cauchy ...

If every Cauchy sequence is convergent , a metric space is said to be

**complete**.The next three lemmas are immediate consequences of the definitions . 6

LEMMA . In a metric space , a convergent sequence is a Cauchy sequence . A

Cauchy ...

Page 311

The space ba ( S , E ) is weakly

) is also weakly

S ) . According to Theorems 6.18 and 6.20 there is a compact Hausdorff space S

...

The space ba ( S , E ) is weakly

**complete**. If S is a topological space , the rba ( S) is also weakly

**complete**. Proof . Consider the closed subspace B ( S , E ) of B (S ) . According to Theorems 6.18 and 6.20 there is a compact Hausdorff space S

...

Page 840

... ( 43 )

compact , 1.6.15 ( 22 ) definition , 1.6.5 ( 19 ) properties , 1.6.7 ( 20 ) , 1.6.9 ( 20 )

Hilbert ...

... ( 43 )

**Complete**and o -**complete**lattice , ( 43 )**Complete**metric space ,compact , 1.6.15 ( 22 ) definition , 1.6.5 ( 19 ) properties , 1.6.7 ( 20 ) , 1.6.9 ( 20 )

**Complete**normed linear space . ( See B - space )**Complete**orthonormal set , inHilbert ...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero