## Linear Operators: General theory |

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Page 97

The total variation v(fi) of an additive set function ft is important because it

dominates // in the sense that v(ft, E) 22 \

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 it

dominates // in the sense that v(ft, E) 22 \

**fi**(**E**)\ for E e E; the reader should test hiscomprehension of Definition 4 below by proving that v(/t) is the smallest of the ...

Page 99

where F is restricted to the domain E of p. The set functions fi~ are additive, non-

negative, and for each E in E,

F Q E, E, F e E then I^F) = li{F)+fi{E)-fi{E-F) <Z p(E)+\n(F)\+\p(E-F)\ <

where F is restricted to the domain E of p. The set functions fi~ are additive, non-

negative, and for each E in E,

**fi**(**E**) = ^(E)-^(E), v(p, E) = fi+(E)+fi-(E). Proof. IfF Q E, E, F e E then I^F) = li{F)+fi{E)-fi{E-F) <Z p(E)+\n(F)\+\p(E-F)\ <

**fi**(**E**)+v(p, E) ...Page 157

Thus the map

subset of M(S,

into Cauchy sequences. Thus if g(

Thus the map

**E**-> %**E**may be regarded as a homeomorphism of**E**(**fi**) onto asubset of M(S,

**E**,**fi**). It is clear that this homeomorphism maps Cauchy sequencesinto Cauchy sequences. Thus if g(

**E**„, Em) 0, there is, by Corollary 6.5, a function ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

79 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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