## Linear Operators: General theory |

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Page 3

The characteristic function iE of a set E is the real function defined by the

equations %E{s) = 1, s e E, and Xe(s) = 0> s 4 E. ... By the extended

system we mean the

the ...

The characteristic function iE of a set E is the real function defined by the

equations %E{s) = 1, s e E, and Xe(s) = 0> s 4 E. ... By the extended

**real number**system we mean the

**real numbers**with the symbols +co and — oo adjoined; bythe ...

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If <f> is the void subset of the

</>, + oo = inf <f>. If A is an infinite set of

denotes the infimum of all numbers b with the property that only a finite set of ...

If <f> is the void subset of the

**real numbers**, it is conventional to take — co = sup</>, + oo = inf <f>. If A is an infinite set of

**real numbers**, then the symbol lim sup Adenotes the infimum of all numbers b with the property that only a finite set of ...

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A subbase for this topology is given by all infinite intervals (— oo, a), (b, +oo),

where a and b are either real or rational. The topology of the extended

A subbase for this topology is given by all infinite intervals (— oo, a), (b, +oo),

where a and b are either real or rational. The topology of the extended

**real****numbers**has as a base the sets [— oo, a), (a,b), (b, +co], where a and b are**real****numbers**...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

80 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

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