## Linear Operators: General theory |

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Page 415

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

of the group G is not essential to the

Abelian topological groups . 4 LEMMA . For arbitrary sets A , B in a linear space X

: ( i ) ...

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

of the group G is not essential to the

**proof**, the same result holds for non -Abelian topological groups . 4 LEMMA . For arbitrary sets A , B in a linear space X

: ( i ) ...

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 = CPx = DPx + De Cx . The

implication CPx + DPK which is the

of 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 = CPx = DPx + De Cx . The

**proof**of theimplication CPx + DPK which is the

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

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero