## Linear Operators: General theory |

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Page 415

Q.E.D. Since the commutativity of the group G is not essential to the

same result holds for non-Abelian topological groups. 4 Lemma. For arbitrary

sets A, B in a linear space £: (i) co(a^) = olco(A), co(A+B) = co(^)+co(B). If £ is a

linear ...

Q.E.D. Since the commutativity of the group G is not essential to the

**proof**, thesame result holds for non-Abelian topological groups. 4 Lemma. For arbitrary

sets A, B in a linear space £: (i) co(a^) = olco(A), co(A+B) = co(^)+co(B). If £ is a

linear ...

Page 434

and proceed as in the first part of the

a subsequence {ym} of {x„} such that lim,,,.^ 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 {x„} such that lim,,,.^ x*ym exists for each x* in the set H of

that

**proof**. Let Km = co{ym, ym+v. . □} and let y0 be an arbitrary point in O Km.Page 699

Therefore dx which proves (*) and completes the

shall now state and prove the lemma referred to as CPk. For technical reasons

occurring later the following lemma is stated for what might be called a positive ...

Therefore dx which proves (*) and completes the

**proof**of the lemma. Q.E.D. Weshall now state and prove the lemma referred to as CPk. For technical reasons

occurring later the following lemma is stated for what might be called a positive ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

79 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

a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero