## Linear Operators: General theory |

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Page 306

According to the Radon-Nikodym theorem (III.10.2) the formula u(E) = jEf(t)X(ds)

establishes an isometric isomorphism between

the set A' in LX(S, E, X) corresponding to A is sequentially weakly compact.

According to the Radon-Nikodym theorem (III.10.2) the formula u(E) = jEf(t)X(ds)

establishes an isometric isomorphism between

**ca**(**S**, E, X) and L^S, E, X). Thusthe set A' in LX(S, E, X) corresponding to A is sequentially weakly compact.

Page 308

establishes an equivalence between

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

hypothesis of Theorem 2, 3 may be chosen so that 31(E) ^ sup \fi(E)\, Eel. Proof.

In view of ...

establishes an equivalence between

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

hypothesis of Theorem 2, 3 may be chosen so that 31(E) ^ sup \fi(E)\, Eel. Proof.

In view of ...

Page 499

1.5, we have, for each E in E, \x*(E)\ = sup \x*(E)x\ S sup I" \{Tx)(*)\v(ft, ds) = |T| =

supu(a:*(-)a;,5) |e|£l iS 4 sup sup \x*(E)x\ = 4 sup |z|Sl EtZ EeS ... 7), and

Theorem 111.2.20(a), the space

1.5, we have, for each E in E, \x*(E)\ = sup \x*(E)x\ S sup I" \{Tx)(*)\v(ft, ds) = |T| =

supu(a:*(-)a;,5) |e|£l iS 4 sup sup \x*(E)x\ = 4 sup |z|Sl EtZ EeS ... 7), and

Theorem 111.2.20(a), the space

**ca**(**S**,E,/i) is equivalent to the space L^S, E, p.).### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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### 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

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 convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set 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 Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero