## Linear Operators: General theory |

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23

additive and bounded.

Show that E* contains every Borel set. 24

topological ...

23

**Suppose**that S is a metric space, that E is a a-field, and that fi is countablyadditive and bounded.

**Suppose**that every continuous function is ^-measurable.Show that E* contains every Borel set. 24

**Suppose**that S is a compacttopological ...

Page 225

Let £7 be a bounded open set in the complex plane, and let B denote the

boundary of 17. We

rectifiable Jordan curves; that is, we

union ...

Let £7 be a bounded open set in the complex plane, and let B denote the

boundary of 17. We

**suppose**that B consists of a finite collection of disjoint closedrectifiable Jordan curves; that is, we

**suppose**that B can be decomposed into theunion ...

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22

finite constant K such that for / in CBV, \(SJ)(x)\ £K(v(f, [0, 2tz])+ sup \f(x)\), 0 £ x ^

2.t. 23

convergence ...

22

**Suppose**that (S„f)(x) f(x) uniformly for every / in AC. Show that there exists afinite constant K such that for / in CBV, \(SJ)(x)\ £K(v(f, [0, 2tz])+ sup \f(x)\), 0 £ x ^

2.t. 23

**Suppose**that (i) (S„f)(x) -+f{x) uniformly in x for / in AC. (ii) Theconvergence ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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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