## Linear Operators: General theory |

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Page 95

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

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

/ a vector (or scalar)

defined ...

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

customarily where integration is used in this text p 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

real-

...

(b) the

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

**function**/ (vector or extendedreal-

**valued**) which is defined only on the complement of a //-null set .V Q S. Then...

Page 196

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

theory of vector

F is a ^-measurable

each ...

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

theory of vector

**valued**integrals. ... Suppose that (S, Z, fi) is a measure space andF is a ^-measurable

**function**whose values are in LP(T, ZT, A), 1 5* p < oo. Foreach ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

79 other sections not shown

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### 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