Linear Operators: General theory |
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Page 168
... separable subset of L ( S , 2 , u , X ) , where 1 ≤ p < . Then there is a set S in 2 , a sub o - field Σ1 of Σ ( S1 ) , and a closed separable subspace X of X such that the restriction μ1 of μ to Σ has the following properties : ( i ) ...
... separable subset of L ( S , 2 , u , X ) , where 1 ≤ p < . Then there is a set S in 2 , a sub o - field Σ1 of Σ ( S1 ) , and a closed separable subspace X of X 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 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
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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 compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality 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 o-field 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 theory topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ