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Page 77
... converges . 48 Show that 00 ∞ ( sin nh \ * Σαπ = lim Σαπ nh n = 1 h → 0 n = 1 whenever k > 1 and the series on the left converges . 49 ( Schur - Mertens ) . Let a = { a } and b quences of complex numbers , and let cm = = { b } be two ...
... converges . 48 Show that 00 ∞ ( sin nh \ * Σαπ = lim Σαπ nh n = 1 h → 0 n = 1 whenever k > 1 and the series on the left converges . 49 ( Schur - Mertens ) . Let a = { a } and b quences of complex numbers , and let cm = = { b } be two ...
Page 145
... converges u - uniformly if for each ε > 0 there is a set E e Σ such that v ( u , E ) < & and such that { f } converges uniformly on S - E . The sequence { f } converges μ - uniformly to the function f if for each & > 0 there is a set E ...
... converges u - uniformly if for each ε > 0 there is a set E e Σ such that v ( u , E ) < & and such that { f } converges uniformly on S - E . The sequence { f } converges μ - uniformly to the function f if for each & > 0 there is a set E ...
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
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ