## Linear Operators: General theory |

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Page 306

According to the Radon-Nikodym theorem (III.10.2) the formula p(E) = jsf(s)X(ds)

establishes an isometric isomorphism between

the set K' in L^S, E, X) corresponding to K is sequentially weakly compact.

According to the Radon-Nikodym theorem (III.10.2) the formula p(E) = jsf(s)X(ds)

establishes an isometric isomorphism between

**ca**(**S**, E, X) and Li(S, E, X). Thusthe set K' in L^S, E, X) corresponding to K 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, X may be chosen so that X(E) ^ sup \fi(E)\, EeZ. Proof.

In view of ...

establishes an equivalence between

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

hypothesis of Theorem 2, X may be chosen so that X(E) ^ sup \fi(E)\, EeZ. Proof.

In view of ...

Page 499

1.5, we have, for each E in Z, \x*{E)\ = sup \x*(E)x\ -g sup (" \(Tx)(8)\v(fi, ds) = \T\ =

sup v(x*(-)x, S) Nisi fg 4 sup sup |a;*(£)-c| = 4 sup JS) [. |a-|Sl Ei£ E(£ ... 7), and

Theorem 111.2.20(a), the space

1.5, we have, for each E in Z, \x*{E)\ = sup \x*(E)x\ -g sup (" \(Tx)(8)\v(fi, ds) = \T\ =

sup v(x*(-)x, S) Nisi fg 4 sup sup |a;*(£)-c| = 4 sup JS) [. |a-|Sl Ei£ E(£ ... 7), and

Theorem 111.2.20(a), the space

**ca**(**S**,Z,fi) is equivalent to the space L^S, Z, fi).### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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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