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Page 77
... converges . 48 Show that Σαπ 00 - Σαn = lim Σ n = 1 h → 0 n = 1 ' sin nh \ nh whenever k > 1 and the series on the left converges . m 49 ( Schur - Mertens ) . Let a = { a } and b = { b } be two se- quences of complex numbers , and let ...
... converges . 48 Show that Σαπ 00 - Σαn = lim Σ n = 1 h → 0 n = 1 ' sin nh \ nh whenever k > 1 and the series on the left converges . m 49 ( Schur - Mertens ) . Let a = { a } and b = { b } be two se- quences of complex numbers , and let ...
Page 145
... converges μ - uniformly if for each ɛ > 0 there is a set E € Σ such that vu , E ) < & and such that { f } converges uniformly on S - E . The sequence { f } converges u - uniformly to the function f if for each & > 0 there is a set E e Σ ...
... converges μ - uniformly if for each ɛ > 0 there is a set E € Σ such that vu , E ) < & and such that { f } converges uniformly on S - E . The sequence { f } converges u - uniformly to the function f if for each & > 0 there is a set E e Σ ...
Page 281
... converges at every point of A to a continuous limit fo . Then { f } converges to fo at every point of S if and only if { f } and every subsequence of { f } converges to fo quasi - uniformly on A. PROOF . Theorem 11 implies that the ...
... converges at every point of A to a continuous limit fo . Then { f } converges to fo at every point of S if and only if { f } and every subsequence of { f } converges to fo quasi - uniformly on A. PROOF . Theorem 11 implies that the ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ