## Linear Operators: General theory |

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Page 3

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

equations Xe(s) = 1> * € 2?, and ££(s) = 0, s 4 E. ... By the extended

system we mean the

the ...

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

equations Xe(s) = 1> * € 2?, and ££(s) = 0, s 4 E. ... By the extended

**real number**system we mean the

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

Page 4

If <f> is the void subset of the

<f>, + 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 — oo = sup<f>, + 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 ...

Page 11

A subbase for this topology is given by all infinite intervals (-co, 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 (-co, 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 [-co, a), (a,b), (b, + oo], 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 | 34 |

Copyright | |

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### Common terms and phrases

a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact