## Linear Operators: General theory |

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Page 106

The

on S are the

every E in E with v(fi, E) < oo, the product %Ef of / with the characteristic

The

**functions**totally u-**measurable**on S, or, if u is understood, totally**measurable**on S are the

**functions**in the closure TM(S) in F(S) of the //-simple**functions**. If forevery E in E with v(fi, E) < oo, the product %Ef of / with the characteristic

**function**...Page 119

(b) the

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

say ...

(b) the

**function**g defined by g{s) = /(«) if s 4 S+ u S~, g{s) = 0 if seS+(jS~> 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

say ...

Page 180

Since every real

(ds), E e E, for a real

a ...

Since every real

**measurable function**is the difference of two non- negative**measurable functions**, it follows from the theorem that [*] j£ g{s)X(ds) = j£ f(s)g(s)fi(ds), E e E, for a real

**measurable function**g. Since the real and imaginary parts ofa ...

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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