Linear Operators: General theory |
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Page 168
... separable subset of L ( S , E , μ , X ) , where 1 ≤ p < ∞o . Then there is a set S1 in Σ , a sub o - field Σ1 of Σ ( S1 ) , and a closed separable subspace X1 of such that the restriction μ1 of μ to Σ1 has the following properties ...
... separable subset of L ( S , E , μ , X ) , where 1 ≤ p < ∞o . Then there is a set S1 in Σ , a sub o - field Σ1 of Σ ( S1 ) , and a closed separable subspace X1 of such that the restriction μ1 of μ to Σ1 has the following properties ...
Page 426
... separable , let { x } be a countable dense subset of X , and define Q ( x * , y * ) = 00 1 | ( x * -y * ) xn \ n1 2 ... separable , and X1 from Corollary II.3.13 that 1 = X. Q.E.D. 2 THEOREM . If X is a B - space , then the X * topology ...
... separable , let { x } be a countable dense subset of X , and define Q ( x * , y * ) = 00 1 | ( x * -y * ) xn \ n1 2 ... separable , and X1 from Corollary II.3.13 that 1 = X. Q.E.D. 2 THEOREM . If X is a B - space , then the X * topology ...
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
Preliminary Concepts | 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 ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ