## Linear Operators: General theory |

### From inside the book

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Page 838

space of,

Arzela theorem, on continuity of limit function, IV.6.11 (268) remarks concerning, (

383) Ascoli-Arzela theorem, on compactness of continuous functions, IV.6.7 ...

space of,

**definition**, IV.2.24 (242) properties, IV.15 Annihilator of a set, II.4.17 (72)Arzela theorem, on continuity of limit function, IV.6.11 (268) remarks concerning, (

383) Ascoli-Arzela theorem, on compactness of continuous functions, IV.6.7 ...

Page 840

Closed orthonormal system ,

set ,

1 ( 70 ) Closed unit sphere , II . 3 . 1 ( 59 ) Closure of a set , criterion to be in , 1 .

Closed orthonormal system ,

**definition**, IV . 14 . 1 ( 357 ) study of , IV . 14 Closedset ,

**definition**, 1 . 4 . 3 ( 10 ) properties , 1 . 4 . 4 - 5 ( 10 ) Closed sphere , 11 . 4 .1 ( 70 ) Closed unit sphere , II . 3 . 1 ( 59 ) Closure of a set , criterion to be in , 1 .

Page 844

11 ( 100 - 101 ) Essential singularity ,

17 ( 23 ) ...

11 ( 100 - 101 ) Essential singularity ,

**definition**, ( 229 ) Essential supremum ,**definition**, III . 1 . 11 ( 100 - 101 ) ... 3 Extended real and complex numbers ,**definitions**, ( 3 ) topology of , ( 11 ) Extension of a function , by continuity , 1 . 6 .17 ( 23 ) ...

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

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