## Linear Operators: General theory |

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Page 164

Since the extension of a

follows that {jin( E)} is a bounded non-decreasing set of real numbers for each E

e Ex. We define ?.X(E) = Iimn fin(E), E e Ev By Corollary 4, Xx is countably ...

Since the extension of a

**non**-**negative**set function on E to Ex is**non**-**negative**, itfollows that {jin( E)} is a bounded non-decreasing set of real numbers for each E

e Ex. We define ?.X(E) = Iimn fin(E), E e Ev By Corollary 4, Xx is countably ...

Page 179

Let (S, Z, ft) be a positive measure space, f a

function defined on S and k(E)=jEf(s)p(ds), EeZ. Let g be a

measurable function defined on S. Then fg is u-meamtrablc, and J£g(S)A(d*) =

fEf(s)g(s)ft(ds) ...

Let (S, Z, ft) be a positive measure space, f a

**non**-**negative**ii-measurablefunction defined on S and k(E)=jEf(s)p(ds), EeZ. Let g be a

**non**-**negative**X-measurable function defined on S. Then fg is u-meamtrablc, and J£g(S)A(d*) =

fEf(s)g(s)ft(ds) ...

Page 314

A subset K of ba(S, 27) is weakly sequentially compact if and only if there exists a

K Q ba(S, 27) be weakly sequentially compact and let V = UT be the isometric ...

A subset K of ba(S, 27) is weakly sequentially compact if and only if there exists a

**non**-**negative**/< in ba(S, 27) such that lim k(E) = 0 uniformly for A e K. Proof. LetK Q ba(S, 27) be weakly sequentially compact and let V = UT be the isometric ...

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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