## Linear Operators: General theory |

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Page 164

Since the extension of a

follows that {//„(£)} is abounded non-decreasing set of real numbers for each E e

Ev We define X1{E) = limn fin(E), E e Ev By Corollary 4, X± is countably additive

on ...

Since the extension of a

**non**-**negative**set function on2'to2'1 is**non**-**negative**, itfollows that {//„(£)} is abounded non-decreasing set of real numbers for each E e

Ev We define X1{E) = limn fin(E), E e Ev By Corollary 4, X± is countably additive

on ...

Page 179

Let (S, E, fi) be a positive measure space, f a

function defined on S and KE)=jEf(s)p(dt), EeE. Let g be a

measurable function defined on S. Then fg is [i-measurablc, and jB g(*Mds) = jE f(

s)g(s)Mds), ...

Let (S, E, fi) be a positive measure space, f a

**non**-**negative**fi- measurablefunction defined on S and KE)=jEf(s)p(dt), EeE. Let g be a

**non**-**negative**X-measurable function defined on S. Then fg is [i-measurablc, and jB g(*Mds) = jE f(

s)g(s)Mds), ...

Page 314

Q.E.D. 12 Theorem. A subset K of ba(S, E) is weakly sequentially compact if and

only if there exists a

XeK. Proof. Let K Q ba(S, E) be weakly sequentially compact and let V = UT be ...

Q.E.D. 12 Theorem. A subset K of ba(S, E) is weakly sequentially compact if and

only if there exists a

**non**-**negative**p in ba(S, E) such that lim X(E) = 0 uniformly forXeK. Proof. Let K Q ba(S, E) be weakly sequentially compact and let V = UT be ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

80 other sections not shown

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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