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 2 , a sub o - field E1 of E ( S1 ) , and a closed separable subspace X1 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 2 , a sub o - field E1 of E ( S1 ) , and a closed separable subspace X1 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 , u ) . 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 , u ) . 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 u - 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 u - 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
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ