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Page 28
... exists a do e D , such that o ( f ( p ) , f ( q ) ) < ɛ if p≥ do , q≥ do . 5 LEMMA . If f is a generalized Cauchy sequence in a complete metric space X , there exists a pe X such that lim f ( d ) - p . PROOF . Let de D be such that c1 ...
... exists a do e D , such that o ( f ( p ) , f ( q ) ) < ɛ if p≥ do , q≥ do . 5 LEMMA . If f is a generalized Cauchy sequence in a complete metric space X , there exists a pe X such that lim f ( d ) - p . PROOF . Let de D be such that c1 ...
Page 353
... exists . Putting 0 < -x If = = lub f ( x ) , show that M1 is a B - space . 1 85 Let K ( x , y ) be a λ × λ ... exists for all 0 < A < ∞ ; ( c ) For each ɛ > 0 and A > 0 there exists a > 0 and N > 0 such that SEK ( x , y ) dy | < ɛ if EC ...
... exists . Putting 0 < -x If = = lub f ( x ) , show that M1 is a B - space . 1 85 Let K ( x , y ) be a λ × λ ... exists for all 0 < A < ∞ ; ( c ) For each ɛ > 0 and A > 0 there exists a > 0 and N > 0 such that SEK ( x , y ) dy | < ɛ if EC ...
Page 362
... exists a complex - valued regular measure u in C * with a1 = n ( y ) u ( dy ) if and only if fx_ _ ∞ Âmnan ‡ n ( x ) \ dx < K , for all m ≥ 1. Show that there exists an ƒ in L , with a , = ( y ) f ( y ) dy if and only if in addition ...
... exists a complex - valued regular measure u in C * with a1 = n ( y ) u ( dy ) if and only if fx_ _ ∞ Âmnan ‡ n ( x ) \ dx < K , for all m ≥ 1. Show that there exists an ƒ in L , with a , = ( y ) f ( y ) dy if and only if in addition ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ