## Linear Operators: General theory |

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Page 140

An

the forms : [ a , b ] = { s a Ss sb } , ( a , b ) = { sa Ss < 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 Ss sb } , ( a , b ) = { sa Ss < 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

i = ni + 1 The argument may be repeated by defining Ez = an , -c and choosing

appropriate points in the

every integer k = 1 , 2 , ... , there are points a ;, bi with c < an Sbn S ... Sabi Sc + &

such ...

i = ni + 1 The argument may be repeated by defining Ez = an , -c and choosing

appropriate points in the

**interval**( c , c + ɛz ] . By induction , it is clear that forevery integer k = 1 , 2 , ... , there are points a ;, bi with c < an Sbn S ... Sabi Sc + &

such ...

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

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero