## Linear Operators: General theory |

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Page 95

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

customarily where integration is used in this text fi is a scalar

/ a vector (or scalar)

defined ...

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

customarily where integration is used in this text fi is a scalar

**valued function**and/ a vector (or scalar)

**valued function**. Thus, even if the integration process isdefined ...

Page 119

(b) the

measurable. Next suppose that we consider a

(b) the

**function**g defined by g(s) = f(s) if s 4 S+ (J S~, g{s) = 0 if s e S+ ij S~, is //-measurable. Next suppose that we consider a

**function**/ (vector or extended real-**valued**) which is defined only on the complement of a /i-null set N C S. Then we ...Page 196

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

theory of vector

and F is a /^-measurable

For each ...

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

theory of vector

**valued**integrals. ... Suppose that (S, E, /u) is a measure spaceand F is a /^-measurable

**function**whose values are in LV(T, ET, A), 1 ^ p < oo.For each ...

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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 compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional 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 Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact