## Linear Operators, Part 1 |

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Page 3

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

equations Xe(s) = 1, se E, and Xe(s) = 0, s # E. ... By the eatended

system we mean the

the ...

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

equations Xe(s) = 1, se E, and Xe(s) = 0, s # E. ... By the eatended

**real number**system we mean the

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

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If q is the void subset of the

+oo = infob. If A is an infinite set of

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

numbers ...

If q is the void subset of the

**real numbers**, it is conventional to take —oo = sup b,+oo = infob. 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

numbers ...

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which has as a base all open intervals (a, b), where a and b are arbitrary

rational numbers. A subbase for this topology is given by all infinite intervals (– co

, a), (b, ...

which has as a base all open intervals (a, b), where a and b are arbitrary

**real****numbers**. Another base for this topology is obtained by taking a and b to berational numbers. A subbase for this topology is given by all infinite intervals (– co

, a), (b, ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero