## 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

f a vector (or scalar)

defined ...

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 isdefined ...

Page 119

(b) the

measurable. Next suppose that we consider a

say ...

(b) the

**function**g defined by g(s) = f(s) if s 6 St. US-, g(s) = 0 if s e S+ US-, is u-measurable. Next suppose that we consider a

**function**f (vector or extended real-**valued**) which is defined only on the complement of a u-null set NC S. Then wesay ...

Page 196

Next we study the relation between the theory of product measures and the

theory of vector

and F is a p-measurable

each s ...

Next we study the relation between the theory of product measures and the

theory of vector

**valued**integrals. ... Suppose that (S, 2, u) is a measure spaceand F is a p-measurable

**function**whose values are in Lp(T, Xr, Ž), 1 < p < 00. Foreach s ...

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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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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