## Linear Operators, Part 1 |

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Page 143

The measure of Lebesgue in an interval [ a , b ] of real numbers may be defined

as the Lebesgue extension of the

measurable sets in [ a , b ] are the sets in the Lebesgue extension ( relative to

The measure of Lebesgue in an interval [ a , b ] of real numbers may be defined

as the Lebesgue extension of the

**Borel**measure in [ a , b ] . The Lebesguemeasurable sets in [ a , b ] are the sets in the Lebesgue extension ( relative to

**Borel**...Page 838

10 (137)

13.8 (223)

partially ordered set, 1.2.3 (4) in the (extended) real number system, (3)

Boundary, ...

10 (137)

**Borel**measure (or**Borel**-Lebesgue measure), construction of, (139) III.13.8 (223)

**Borel**-Stieltjes measure, (142) Bound, of an operator, II.3.5 (60) in apartially ordered set, 1.2.3 (4) in the (extended) real number system, (3)

Boundary, ...

Page 838

1 ( 41 )

Lebesgue measure ) , construction of , ( 139 ) III . 13 . 8 ( 223 )

measure , ( 142 ) Bound , of an operator , II . 3 . 5 ( 60 ) in a partially ordered set ,

1 ...

1 ( 41 )

**Borel**field of sets , definition , I11 . 5 . 10 ( 137 )**Borel**measure ( or**Borel**-Lebesgue measure ) , construction of , ( 139 ) III . 13 . 8 ( 223 )

**Borel**- Stieltjesmeasure , ( 142 ) Bound , of an operator , II . 3 . 5 ( 60 ) in a partially ordered set ,

1 ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 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 obtained 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