Linear Operators: General theory |
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Page 177
Nelson Dunford, Jacob T. Schwartz. may be assumed that λ is real valued . A real valued set function can be represented as ... assumed to be positive and measurable we can put F = { ss e F , f ( s ) ≤ n } . Then F = UF , and v ( 2 , F ) ...
Nelson Dunford, Jacob T. Schwartz. may be assumed that λ is real valued . A real valued set function can be represented as ... assumed to be positive and measurable we can put F = { ss e F , f ( s ) ≤ n } . Then F = UF , and v ( 2 , F ) ...
Page 278
... assumed that x * = x * f . Then there is a point s in S such that x * f = f ( s ) , fe C ( S ) . PROOF . By the preceding lemma x * is a point in the space S1 de- fined in the proof of Theorem 18. By Theorem 22 , S is homeomorphic to a ...
... assumed that x * = x * f . Then there is a point s in S such that x * f = f ( s ) , fe C ( S ) . PROOF . By the preceding lemma x * is a point in the space S1 de- fined in the proof of Theorem 18. By Theorem 22 , S is homeomorphic to a ...
Page 675
... assumed that ƒ is in L1 . In view of Lemma 4 it may also be assumed that T is positive . Let e∞ = { s sup A ( T , k ) ( f , s ) 1≤k = ∞0 } . Since μ is o - finite on S it will suffice to prove that μ ( ee ) = 0 for every set e in Σ ...
... assumed that ƒ is in L1 . In view of Lemma 4 it may also be assumed that T is positive . Let e∞ = { s sup A ( T , k ) ( f , s ) 1≤k = ∞0 } . Since μ is o - finite on S it will suffice to prove that μ ( ee ) = 0 for every set e in Σ ...
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