Linear Operators: General theory |
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Page 212
... positive measure defined on the o - field of Borel sets of a compact metric space S. A set ACS is said to be covered in the sense of Vitali by a family F of closed sets if each Fe F has positive μ - measure and there is a positive ...
... positive measure defined on the o - field of Borel sets of a compact metric space S. A set ACS is said to be covered in the sense of Vitali by a family F of closed sets if each Fe F has positive μ - measure and there is a positive ...
Page 304
... positive and write T≥ 0 when ƒ e L , and f≥ 0 imply that Tf ≥0 . Similarly T1 ≥ T , or T2 ≤ T1 means that T1 ... positive measure space and let 1≤poo . Suppose f1 + f2 Σ - 18k , where fi , gx are positive elements = n in L , ( S ...
... positive and write T≥ 0 when ƒ e L , and f≥ 0 imply that Tf ≥0 . Similarly T1 ≥ T , or T2 ≤ T1 means that T1 ... positive measure space and let 1≤poo . Suppose f1 + f2 Σ - 18k , where fi , gx are positive elements = n in L , ( S ...
Page 305
... positive elements . Let ƒ , € L2 ( S , Σ , μ ) , f ; ≥ 0 , j 12 n = 1 , 2 , and let f1 = -1611 ) f2 = Σ - 1821 be decompositions of f1 and f2 into fi positive functions . Then Σg1 + Σgi is a decomposition of ƒ1 + ƒ1⁄2 , and so To ( f1 ...
... positive elements . Let ƒ , € L2 ( S , Σ , μ ) , f ; ≥ 0 , j 12 n = 1 , 2 , and let f1 = -1611 ) f2 = Σ - 1821 be decompositions of f1 and f2 into fi positive functions . Then Σg1 + Σgi is a decomposition of ƒ1 + ƒ1⁄2 , and so To ( f1 ...
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 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ