## Linear Operators: General theory |

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Page 28

is a generalized Cauchy sequence in a complete metric space X, there

a p e A' such that lim f(d) = p. Proof. Let dn e D be such that cv c2 2: d„ implies (?(/

(<?! ), f(c2)) < 1/n. Let 6n be an upper bound for the finite set {dv d2, . . ., dn}. Then

it ...

is a generalized Cauchy sequence in a complete metric space X, there

**exists**a p e A' such that lim f(d) = p. Proof. Let dn e D be such that cv c2 2: d„ implies (?(/

(<?! ), f(c2)) < 1/n. Let 6n be an upper bound for the finite set {dv d2, . . ., dn}. Then

it ...

Page 353

Let M1 be the space of all bounded A-measurable functions / defined on R such

that lim f(x)

measurable function on RxR. Suppose that K(x, y) is A-integrable for each fixed x.

Let M1 be the space of all bounded A-measurable functions / defined on R such

that lim f(x)

**exists**. Putting show that Mt is a B-space. 85 Let K(x, y) be a AxAmeasurable function on RxR. Suppose that K(x, y) is A-integrable for each fixed x.

Page 362

Under the hypotheses of Exercise 37, show that there

)<f>n(x)dx if and only if the functions 2£.-«^m»»an<Ma,)> m S: 1, are uniformly

bounded and equicontinuous. 39 Let {a„}, — oo < « < +oo, be a bounded ...

Under the hypotheses of Exercise 37, show that there

**exists**/ in C with an = So'f(x)<f>n(x)dx if and only if the functions 2£.-«^m»»an<Ma,)> m S: 1, are uniformly

bounded and equicontinuous. 39 Let {a„}, — oo < « < +oo, be a bounded ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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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