Linear Operators: General theory |
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Page 188
... corollary is the o - finite analogue of Corollary 4 . 7 COROLLARY . Let ( S , Z. μ ) be the product of two positive o - finite measure spaces ( §1 , 21 , μ1 ) and ( S2 , 22 , 2 ) . For each E in 2 and 82 in S2 the set E ( 82 ) = { 81 ...
... corollary is the o - finite analogue of Corollary 4 . 7 COROLLARY . Let ( S , Z. μ ) be the product of two positive o - finite measure spaces ( §1 , 21 , μ1 ) and ( S2 , 22 , 2 ) . For each E in 2 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 n x Σ b ...
... 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 n x Σ b ...
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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ