Linear Operators: General theory |
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Page 306
... ca ( S , E , 2 ) consisting of all 2 - continuous functions in ca ( S , E ) . According to the Radon - Nikodým theorem ( III.10.2 ) the formula μ ( E ) = √_f ( s ) λ ( ds ) establishes an isometric isomorphism between ca ( S , E , 2 ) ...
... ca ( S , E , 2 ) consisting of all 2 - continuous functions in ca ( S , E ) . According to the Radon - Nikodým theorem ( III.10.2 ) the formula μ ( E ) = √_f ( s ) λ ( ds ) establishes an isometric isomorphism between ca ( S , E , 2 ) ...
Page 308
... ca ( S , E ) is weakly complete . PROOF . If { u } is a weak Cauchy sequence in ca ( S , Σ ) then the limit lim ( E ) exists for every E in E and , by II.3.27 , the sequence { n } is bounded . According to Corollary III.7.4 the ...
... ca ( S , E ) is weakly complete . PROOF . If { u } is a weak Cauchy sequence in ca ( S , Σ ) then the limit lim ( E ) exists for every E in E and , by II.3.27 , the sequence { n } is bounded . According to Corollary III.7.4 the ...
Page 499
... ( s ) | v ( μ , ds ) | ≤1 = T = | x | ≤1 sup v ( x * ( ) x , S ) x≤1 ≤ 4 sup sup x * ( E ) x | = 4 sup x * ( E ) ... ca ( S , E , μ ) of ca ( S , E ) which consists of all u - continuous func- tions in ca ( S , E ) . By the general Radon ...
... ( s ) | v ( μ , ds ) | ≤1 = T = | x | ≤1 sup v ( x * ( ) x , S ) x≤1 ≤ 4 sup sup x * ( E ) x | = 4 sup x * ( E ) ... ca ( S , E , μ ) of ca ( S , E ) which consists of all u - continuous func- tions in ca ( S , E ) . By the general Radon ...
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