## Linear Operators, Part 1 |

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Page 306

The functions u, are all continuous with respect to the measure defined by 2(e) –

S "ro, Ee 2, ,-, 2n 1 +v(un, E) and thus all belong to the subspace

consisting of all Ż-continuous functions in

Nikodym ...

The functions u, are all continuous with respect to the measure defined by 2(e) –

S "ro, Ee 2, ,-, 2n 1 +v(un, E) and thus all belong to the subspace

**cas**.S. 2, 2)consisting of all Ż-continuous functions in

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

Page 308

establishes an equivalence between

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

hypothesis of Theorem 2, A may be chosen so that A(E) < sup u(E), E e X. puek

PRoof.

establishes an equivalence between

**ca**(**S**, X, A) and L1(S, X, A) and thus thepresent theorem follows from Corollary 8.11. Q.E.D. 3 CoRoll.ARY. Under the

hypothesis of Theorem 2, A may be chosen so that A(E) < sup u(E), E e X. puek

PRoof.

Page 499

(T,)(s) (u, ds) |a = 1 |z| <1 = |T| = sup v(r”(-)r, S) |als 1 < 4 sup sup w”(E)r = 4 sup a

“(E). |a|<1 Ee: Eex ... By the general Radon-Nikodym theorem (III.10.7), and

Theorem III.2.20(a), the space

(T,)(s) (u, ds) |a = 1 |z| <1 = |T| = sup v(r”(-)r, S) |als 1 < 4 sup sup w”(E)r = 4 sup a

“(E). |a|<1 Ee: Eex ... By the general Radon-Nikodym theorem (III.10.7), and

Theorem III.2.20(a), the space

**ca**(**S**, 2, u) is equivalent to the space L1(S, 2, u).### What people are saying - Write a review

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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