## Linear Operators: General theory |

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Page 97

The total variation

dominates // in the sense that v(ft, E) 22 \fi(E)\ for E e E; the reader should test his

comprehension of Definition 4 below by proving that v(/t) is the smallest of the ...

The total variation

**v**(**fi**) of an additive set function ft is important because itdominates // in the sense that v(ft, E) 22 \fi(E)\ for E e E; the reader should test his

comprehension of Definition 4 below by proving that v(/t) is the smallest of the ...

Page 98

Then E\p(Et)\ £S E^F^ and we have v(E) = lira 2>(-E<)|- 6 Lemma. The total

variation of an additive set ... n and hence (i)

E\j F) = oo it follows that v(p, E\jF) = i>(/i, F) + u(/<, F). If £u F) < oo there are finite ...

Then E\p(Et)\ £S E^F^ and we have v(E) = lira 2>(-E<)|- 6 Lemma. The total

variation of an additive set ... n and hence (i)

**v**(**fi**, EuF) ^ v(ft, E)+**v**(**fi**, F). Thus if**v**(**fi**,E\j F) = oo it follows that v(p, E\jF) = i>(/i, F) + u(/<, F). If £u F) < oo there are finite ...

Page 157

The space E(fi) is therefore a complete metric space. If X is an additive vector or

scalar valued function on E which is ^-continuous, then X is defined and

continuous on the metric space E{fi). To see this note first that the identities

EA F) ...

The space E(fi) is therefore a complete metric space. If X is an additive vector or

scalar valued function on E which is ^-continuous, then X is defined and

continuous on the metric space E{fi). To see this note first that the identities

**v**(**fi**,EA F) ...

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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