## Linear Operators: General theory |

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Page 306

The functions [in are all continuous with respect to the measure defined by • 1 v(u

„, E) k(E)=y ^" ' , Eel and thus all belong to the subspace

all A-continuous functions in

...

The functions [in are all continuous with respect to the measure defined by • 1 v(u

„, E) k(E)=y ^" ' , Eel and thus all belong to the subspace

**ca**(**S**, E, A) consisting ofall A-continuous functions in

**ca**(**S**, E). According to the Radon-Nikodym theorem...

Page 308

establishes an equivalence between

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

hypothesis of Tlieorem 2, A may be chosen so that litK Proof. In view of Lemma III.

1.5 and ...

establishes an equivalence between

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

hypothesis of Tlieorem 2, A may be chosen so that litK Proof. In view of Lemma III.

1.5 and ...

Page 499

1.5, we have, for eaeh E in E, \x*{E)\ = sup \x*(E)x\ g sup f \(Tx)(s)\v(p, ds) = |Tj =

supu(a:*(-)*, S) g 4 sup sup \x*(E)x\ = 4 sup \x*(E)\. Conversely, if x*{-) ... 10.7).

and Theorem 111.2.20(a), the space

p).

1.5, we have, for eaeh E in E, \x*{E)\ = sup \x*(E)x\ g sup f \(Tx)(s)\v(p, ds) = |Tj =

supu(a:*(-)*, S) g 4 sup sup \x*(E)x\ = 4 sup \x*(E)\. Conversely, if x*{-) ... 10.7).

and Theorem 111.2.20(a), the space

**ca**(**S**,E,p) is equivalent to the space L^S. E,p).

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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 compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional 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 Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact