## Linear Operators: General theory |

### From inside the book

Results 1-3 of 86

Page 106

The

on S are the

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

The

**functions**totally fi-measurahle on S, or, if // 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(/i, E) < oo, the product %Ff of / with the characteristic

**function**...Page 179

4 Theorem. Let (S, E, a ) be a positive measure space, f a non- negative ju-

negative k-

s)l(<ls) ...

4 Theorem. Let (S, E, a ) be a positive measure space, f a non- negative ju-

**measurable function**defined on S and X(E)=jEf(s)Mds), Eel. Let g be a non-negative k-

**measurable function**defined on S. Then fg is f i- measurable, and j£ g(s)l(<ls) ...

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 [*] jE g(s)X(ds) = j£ f(s)g(s)/i(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 | |

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