## Linear Operators: General theory |

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Page 95

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

customarily where integration is used in this text [x is a scalar

/ a vector (or scalar)

defined ...

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

customarily where integration is used in this text [x 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+ (J S~, is //-measurable. Next suppose that we consider a

**function**/ (vector or extended real-**valued**) which is defined only on the complement of a //-null set N Q 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, fi) is a measure spaceand F is a /^-measurable

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

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

44 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

Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma 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 properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero