## Linear Operators, Part 1 |

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Page 95

In some cases that will be encountered the values of u are not scalars, but

customarily where integration is used in this text u is a scalar valued

f a vector (or scalar) valued

In some cases that will be encountered the values of u are not scalars, but

customarily where integration is used in this text u is a scalar valued

**function**andf a vector (or scalar) valued

**function**. Thus, even if the integration process is**defined**...Page 98

The total variation of an additive set

set S is also additive on 2. PRoof. Let {A,} be a finite sequence of disjoint sets in 2

with A, C E U F where E, F e 2 and EF = d. Let E, - EA, F, - FA, then Xu (A)| : X ...

The total variation of an additive set

**function defined**on a field 2 of subsets of aset S is also additive on 2. PRoof. Let {A,} be a finite sequence of disjoint sets in 2

with A, C E U F where E, F e 2 and EF = d. Let E, - EA, F, - FA, then Xu (A)| : X ...

Page 101

In these sections f will be a vector valued

finitely additive set

assumed that u is bounded. Thus the basis for this material is a fixed set S, a field

2 of ...

In these sections f will be a vector valued

**function defined**on a set S and u afinitely additive set

**function defined**on a field of subsets of S. It will not beassumed that u is bounded. Thus the basis for this material is a fixed set S, a field

2 of ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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### Common terms and phrases

additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero