Linear Operators: General theory |
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Page 188
... corollary is the o - finite analogue of Corollary 4 . 7 COROLLARY . Let ( S , E , u ) be the product of two positive o - finite measure spaces ( $ 1 , 21 , μ1 ) and ( S2 , Z2 , μ2 ) . For each E in Σ and s2 in S2 the set E ( 82 ) ...
... corollary is the o - finite analogue of Corollary 4 . 7 COROLLARY . Let ( S , E , u ) be the product of two positive o - finite measure spaces ( $ 1 , 21 , μ1 ) and ( S2 , Z2 , μ2 ) . For each E in Σ and s2 in S2 the set E ( 82 ) ...
Page 246
... corollary was established during the first part of the preceding proof . 7 COROLLARY . If { b1 , . . . , b2 } is a Hamel basis for the normed linear space X then the functionals b * , i = 1 , . . . , n , defined by the equa- tions n x ...
... corollary was established during the first part of the preceding proof . 7 COROLLARY . If { b1 , . . . , b2 } is a Hamel basis for the normed linear space X then the functionals b * , i = 1 , . . . , n , defined by the equa- tions n x ...
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 ( ) = 0 ; by Theorem 9 , for . Put f = folfo ( x ) , and the corollary is proved . Q.E.D. 13 THEOREM . A convex ...
... 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 ( ) = 0 ; by Theorem 9 , for . Put f = folfo ( x ) , and the corollary is proved . Q.E.D. 13 THEOREM . A convex ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz 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