## Linear Operators: General theory |

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Page 28

f : D -*□ X be a generalized sequence of elements in a metric space X. We call /

a generalized Cauchy sequence in X, if, for each e > 0, there

that Q(f(p), f(q)) < e if p ^ d0, q ^ d0. 5 Lemma. // / is a generalized Cauchy ...

f : D -*□ X be a generalized sequence of elements in a metric space X. We call /

a generalized Cauchy sequence in X, if, for each e > 0, there

**exists**a d0e D, suchthat Q(f(p), f(q)) < e if p ^ d0, q ^ d0. 5 Lemma. // / is a generalized Cauchy ...

Page 362

Under the hypotheses of Exercise 87, show that there

#i»(<»0<fo if and only if the functions »^mn«n^n(«). m ^ 1, are uniformly bounded

and equicontinuous. 39 Let {an}, — oo<n<+oo, be a bounded sequence of ...

Under the hypotheses of Exercise 87, show that there

**exists**/ in C with fln = Jo"/(a#i»(<»0<fo if and only if the functions »^mn«n^n(«). m ^ 1, are uniformly bounded

and equicontinuous. 39 Let {an}, — oo<n<+oo, be a bounded sequence of ...

Page 721

Decompose v into its ^-continuous and fi- singular parts. Using the method of

Banach's theorem IV. 11. 2 and the C(5)-density of Ly(S, 38, fi) in ca{3&), show

that for e>0 there

d.) ...

Decompose v into its ^-continuous and fi- singular parts. Using the method of

Banach's theorem IV. 11. 2 and the C(5)-density of Ly(S, 38, fi) in ca{3&), show

that for e>0 there

**exists**a d > 0 such that H-\ SUP I \Kn{s, t)v(dt) > e) < s if v(v,S) <d.) ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function 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 Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero