Linear Operators: General theory |
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Page 188
... corollary is the o - finite analogue of Corollary 4 . 7 COROLLARY . Let ( S , E , μ ) be the product of two positive o - finite measure spaces ( S1 , 21 , μ1 ) and ( S2 , 22 , 2 ) . For each E in E and 82 in S2 the set E ( 82 ) = { 81 ...
... corollary is the o - finite analogue of Corollary 4 . 7 COROLLARY . Let ( S , E , μ ) be the product of two positive o - finite measure spaces ( S1 , 21 , μ1 ) and ( S2 , 22 , 2 ) . For each E in E and 82 in S2 the set E ( 82 ) = { 81 ...
Page 246
... corollary was established during the first part of the preceding proof . 7 COROLLARY . If { b1 , ... , b ) is a Hamel basis for the normed linear space X then the functionals b * , i = 1 , . . . , n , defined by the equa- tions ก x = Σ ...
... corollary was established during the first part of the preceding proof . 7 COROLLARY . If { b1 , ... , b ) is a Hamel basis for the normed linear space X then the functionals b * , i = 1 , . . . , n , defined by the equa- tions ก x = Σ ...
Page 422
... Corollary 2.12 , there is a T - continuous fo and a constant c such that Rfo ( Y ) ≤ c , fo ( x ) 0. By Lemma 1.11 , fo ( 9 ) = 0 ; by Theorem 9 , foe г. Put ƒ = fo / fo ( a ) , and the corollary is proved . Q.E.D. 13 THEOREM . A ...
... Corollary 2.12 , there is a T - continuous fo and a constant c such that Rfo ( Y ) ≤ c , fo ( x ) 0. By Lemma 1.11 , fo ( 9 ) = 0 ; by Theorem 9 , foe г. Put ƒ = fo / fo ( a ) , and the corollary is proved . Q.E.D. 13 THEOREM . A ...
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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ