## Linear Operators: General theory |

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Page 106

Nelson Dunford, Jacob T. Schwartz. 10 Definition. The

in the closure TM(S) in F(S) of the /i-simple

E) < oo ...

Nelson Dunford, Jacob T. Schwartz. 10 Definition. The

**functions**totally /x-**measurable**on S, or, if fi is understood, totally**measurable**on S are the**functions**in the closure TM(S) in F(S) of the /i-simple

**functions**. If for every E in 27 with v(fi,E) < oo ...

Page 180

Since every real

ds), 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(*Wd») = jEf(s)g(s)v(ds), E € £, for a real

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

Page 663

Since fi(S) < oo, every /^-

III. 2. 10) and thus, as was shown at the beginning of Section I VT.l 1 , the space

M(S) is an F-space. The condition (ii) shows that T maps //-equivalent functions ...

Since fi(S) < oo, every /^-

**measurable function**is totally //-measurable (DefinitionIII. 2. 10) and thus, as was shown at the beginning of Section I VT.l 1 , the space

M(S) is an F-space. The condition (ii) shows that T maps //-equivalent functions ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional 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 Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact