## Linear Operators: General theory |

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Page 164

Let C be the set of all fj. in

chosen from C in such a way that limn^B(<S) = sup^eC n(S) < co. Since pt ^ » = L

• • •> «> it follows from Corollary 6 that {/^ has a least upper bound p.n in

Let C be the set of all fj. in

**ca**(**S**, E) such that 0 :g fi ^ A. Let fin, n = 1, 2, . . ., bechosen from C in such a way that limn^B(<S) = sup^eC n(S) < co. Since pt ^ » = L

• • •> «> it follows from Corollary 6 that {/^ has a least upper bound p.n in

**ca**(**S**, E).Page 306

The functions fi„ are all continuous with respect to the measure defined by and

thus all belong to the subspace

functions in

formula f*(B) ...

The functions fi„ are all continuous with respect to the measure defined by and

thus all belong to the subspace

**ca**(**S**, E, X) consisting of all A-continuousfunctions in

**ca**(**S**, E). According to the Radon-Nikodym theorem (III. 10.2) theformula f*(B) ...

Page 308

establishes an equivalence between

present theorem follows from Corollary 8.11. Q.E.D. 3 Corollary. Under the

hypothesis of Theorem 2, X may he chosen so that X{E) g mp \p(E)\, EeE. IteK

Proof.

establishes an equivalence between

**ca**(**S**, 27, X) and L^S, E, X) and thus thepresent theorem follows from Corollary 8.11. Q.E.D. 3 Corollary. Under the

hypothesis of Theorem 2, X may he chosen so that X{E) g mp \p(E)\, EeE. IteK

Proof.

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

80 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-finite Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space valued function vector space weak topology weakly compact weakly sequentially compact zero