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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 dn e D be ...
... 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 dn e D be ...
Page 353
... exists . Putting 04 * If = lub f ( x ) , ∞ > x50 show that M1 is a B - space . 85 Let K ( x , y ) be a λ × λ ... exists for all 0 < A < ∞o ; ( c ) For each ɛ > 0 and A > 0 there exists a > 0 and N > 0 such that SEK ( x , y ) dy | < ɛ ...
... exists . Putting 04 * If = lub f ( x ) , ∞ > x50 show that M1 is a B - space . 85 Let K ( x , y ) be a λ × λ ... exists for all 0 < A < ∞o ; ( c ) For each ɛ > 0 and A > 0 there exists a > 0 and N > 0 such that SEK ( x , y ) dy | < ɛ ...
Page 362
... exists a complex - valued regular measure μ in C * with an if and only if fΣxx Amna¤n ( x ) \ da < K , for all m ≥ 1. Show that there exists an ƒ in L1 with a , = √ ( y ) f ( y ) dy if and only if in addition to the preceding ...
... exists a complex - valued regular measure μ in C * with an if and only if fΣxx Amna¤n ( x ) \ da < K , for all m ≥ 1. Show that there exists an ƒ in L1 with a , = √ ( y ) f ( y ) dy if and only if in addition to the preceding ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ 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 X₁ zero ΕΕΣ