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 ( S1 , E1 , μ1 ) and ( S2 , Z2 , μ2 ) . For each E in Σ and s2 in S2 the set E ( 82 ) = { $ 1 ...
... 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 ( S1 , E1 , μ1 ) and ( S2 , Z2 , μ2 ) . For each E in Σ and s2 in S2 the set E ( 82 ) = { $ 1 ...
Page 246
... corollary was established during the first part of the preceding proof . 7 COROLLARY . If { b1 , bn } 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 * ( x ) ...
... corollary was established during the first part of the preceding proof . 7 COROLLARY . If { b1 , bn } 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 * ( x ) ...
Page 422
... Corollary 2.12 , there is a T - continuous fo and a constant c such that fo ( Y ) ≤ c , fo ( x ) 0. By Lemma 1.11 , fo ( 9 ) = 0 ; by Theorem 9 , fo € Tг . Put ƒ = fo / fo ( x ) , 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 fo ( Y ) ≤ c , fo ( x ) 0. By Lemma 1.11 , fo ( 9 ) = 0 ; by Theorem 9 , fo € Tг . Put ƒ = fo / fo ( x ) , and the corollary is proved . Q.E.D. 13 THEOREM . A ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ