## Linear Operators: General theory |

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Page 140

An

the forms: [a, b] = {s\a :g * :£ b}, [a, b) — {s\a ^ s <b}, (a, b] = {s\a <s ... The number

a is called the left end point and b the right end point of any of these

An

**interval**is a set of points in the extended real number system which has one ofthe forms: [a, b] = {s\a :g * :£ b}, [a, b) — {s\a ^ s <b}, (a, b] = {s\a <s ... The number

a is called the left end point and b the right end point of any of these

**intervals**.Page 141

It is assumed that (i) /(*) = lim/(« + |e|), «€/; e-*o i.e., / is continuous on the right at

every point in the open

measure, we let 2 be the field of all finite unions (ii) E = IiUhU □ □ >L)In of

It is assumed that (i) /(*) = lim/(« + |e|), «€/; e-*o i.e., / is continuous on the right at

every point in the open

**interval**J. The ... in the above construction of Borelmeasure, we let 2 be the field of all finite unions (ii) E = IiUhU □ □ >L)In of

**intervals**Iit j ...Page 223

5 Let h be a function of bounded variation on the

the right. Let g be a function defined on (a, b) such that the Lebesgue-Stieltjes

integral I = §\g(s)dh(s) exists. Let / be a continuous increasing function on an

open ...

5 Let h be a function of bounded variation on the

**interval**(a, b) and continuous onthe right. Let g be a function defined on (a, b) such that the Lebesgue-Stieltjes

integral I = §\g(s)dh(s) exists. Let / be a continuous increasing function on an

open ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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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