Linear Operators: General theory |
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Page 182
... defined u - almost everywhere by the formula 2 ( E ) == [ { d2 ( 8 ) μ ( ds ) . JE ΕΕΣ . We close this section with ... defined on S2 is μ - measurable , then f ( $ ( · ) ) is μ - measurable ; ( e ) if u is non - negative and countably ...
... defined u - almost everywhere by the formula 2 ( E ) == [ { d2 ( 8 ) μ ( ds ) . JE ΕΕΣ . We close this section with ... defined on S2 is μ - measurable , then f ( $ ( · ) ) is μ - measurable ; ( e ) if u is non - negative and countably ...
Page 240
... defined for a field of subsets of a set S and consists of all bounded additive scalar functions defined on 2. The norm u is the total variation of μ on S , i.e. , | μ | = v ( μ , S ) . 16. The space ca ( S , E ) is defined for a σ ...
... defined for a field of subsets of a set S and consists of all bounded additive scalar functions defined on 2. The norm u is the total variation of μ on S , i.e. , | μ | = v ( μ , S ) . 16. The space ca ( S , E ) is defined for a σ ...
Page 534
... defined by an = 1 % x " f ( x ) dx , n≥0 , is a bounded map of L2 into l2 such that TT * is the map { a } → { b } defined by bn = 00 aj Σ ¿ = 0 n + j +1 Show that T has norm √л . Show that the map S of L2 ( 0 , 1 ) into itself defined ...
... defined by an = 1 % x " f ( x ) dx , n≥0 , is a bounded map of L2 into l2 such that TT * is the map { a } → { b } defined by bn = 00 aj Σ ¿ = 0 n + j +1 Show that T has norm √л . Show that the map S of L2 ( 0 , 1 ) into itself defined ...
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