## Linear Operators: General theory |

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Results 1-3 of 82

Page 103

For an example of such a

of intervals / = [a, b), 0 a < b < 1, with fi(I) = b— a as in Section 1. Let R denote the

set of rational points in S. For r = p;q e R in lowest terms, we define

For an example of such a

**function**, let S = [0, 1 ) and E be the field of finite unionsof intervals / = [a, b), 0 a < b < 1, with fi(I) = b— a as in Section 1. Let R denote the

set of rational points in S. For r = p;q e R in lowest terms, we define

**F**(pjq) = l/q, ...Page 149

If (S, E, fi) is a measure space, then a

measurable if and only if (i) for every measurable set F with v(fi, F) < oo, the

x* in X* the ...

If (S, E, fi) is a measure space, then a

**function f**on S to a B-space X is p-measurable if and only if (i) for every measurable set F with v(fi, F) < oo, the

**function f**is fi-essentially separably valued on F, and (ii) for every linear functionalx* in X* the ...

Page 196

For each s in S, F(s) is an equivalence class of functions, any pair of whose

members coincide A-almost everywhere. If for each s we select a particular

(T,ET,X) ...

For each s in S, F(s) is an equivalence class of functions, any pair of whose

members coincide A-almost everywhere. If for each s we select a particular

**function f**(s, □) e F(s), the resulting**function f**(s, t) defined on [R,ER,e) = (S,Es,M)x(T,ET,X) ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

80 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

a-finite 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 continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space 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 valued function vector space weak topology weakly compact weakly sequentially compact zero