## Linear Operators: General theory |

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Page 415

Hence ko e p - A , and thus p e A + k , CA + K . Q.E.D. Since the commutativity of

the group G is not essential to the

topological groups . 4 LEMMA . For arbitrary sets A , B in a linear space X : ( i ) co

...

Hence ko e p - A , and thus p e A + k , CA + K . Q.E.D. Since the commutativity of

the group G is not essential to the

**proof**, the same result holds for non - Abeliantopological groups . 4 LEMMA . For arbitrary sets A , B in a linear space X : ( i ) co

...

Page 434

and proceed as in the first part of the

a subsequence { ym } of { Xn } such that limm - * x * ym exists for each x * in the

set H of that

...

and proceed as in the first part of the

**proof**of the preceding theorem to constructa subsequence { ym } of { Xn } such that limm - * x * ym exists for each x * in the

set H of that

**proof**. Let Km = co { ym , Ym + ] , . . . } and let yo be an arbitrary point...

Page 699

The

11 as outlined in the diagram : C → CPk DPx = Dx → Cx . The

implication CP DPk which is the

the ...

The

**proof**of this lemma is the most involved of all the steps in the**proof**of Lemma11 as outlined in the diagram : C → CPk DPx = Dx → Cx . The

**proof**of theimplication CP DPk which is the

**proof**of Lemma 14 is very similar to the**proof**ofthe ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Common terms and phrases

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 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