## Linear Operators: General theory |

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Page 140

An

the forms : [ a , b ] = { s a < s < b } , [ a , b ) = { sa Is < b } , ( a , b ] ... 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 < s < b } , [ a , b ) = { sa Is < b } , ( a , b ] ... 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 ) f ( s ) = lim f ( s + el ) , sel ; E - > 0 i . e . , f is continuous on

the right at every point in the open

compact subset of the extended real number system and we extend the domain

off ...

It is assumed that ( i ) f ( s ) = lim f ( s + el ) , sel ; E - > 0 i . e . , f is continuous on

the right at every point in the open

**interval**I . The closed**interval**Ī = [ a , b ] is acompact subset of the extended real number system and we extend the domain

off ...

Page 223

5 Let h be a function of bounded variation on the

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

Stieltjes integral I = Sag ( s ) dh ( s ) exists . Let f be a continuous increasing

function ...

5 Let h be a function of bounded variation on the

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

Stieltjes integral I = Sag ( s ) dh ( s ) exists . Let f be a continuous increasing

function ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero