Linear Operators: General theory |
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Page 168
... separable subset of L ( S , Σ , μ , X ) , where 1 ≤ p < ∞ . Then there is a set S in E , a sub o - field E1 of E ( S1 ) , and a closed separable subspace X of such that the restriction μ1 of μ to Σ has the following properties : ( i ) ...
... separable subset of L ( S , Σ , μ , X ) , where 1 ≤ p < ∞ . Then there is a set S in E , a sub o - field E1 of E ( S1 ) , and a closed separable subspace X of such that the restriction μ1 of μ to Σ has the following properties : ( i ) ...
Page 501
... separable . Q.E.D. Let U be the closed unit sphere in L1 ( S , Σ , μ ) . Then , by the pre- ceding lemma , since K = TU is compact and has a countable base the space C ( K ) is separable . Since a subspace of a separable metric space is ...
... separable . Q.E.D. Let U be the closed unit sphere in L1 ( S , Σ , μ ) . Then , by the pre- ceding lemma , since K = TU is compact and has a countable base the space C ( K ) is separable . Since a subspace of a separable metric space is ...
Page 507
... separable sub- set of the B - space X. Then there exists a μ - essentially unique bounded measurable function x ... separable range . PROOF . Let U be the closed unit sphere of L1 ( S , Σ , μ ) . If T is weakly compact and has a ...
... separable sub- set of the B - space X. Then there exists a μ - essentially unique bounded measurable function x ... separable range . PROOF . Let U be the closed unit sphere of L1 ( S , Σ , μ ) . If T is weakly compact and has a ...
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