Linear Operators: General theory |
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Page 306
... ca ( S , Σ , λ ) consisting of all 2 - continuous functions in ca ( S , E ) . According to the Radon - Nikodým theorem ( III.10.2 ) the formula μ ( E ) = √2f ( s ) 2 ( ds ) E establishes an isometric isomorphism between ca ( S , E , λ ) ...
... ca ( S , Σ , λ ) consisting of all 2 - continuous functions in ca ( S , E ) . According to the Radon - Nikodým theorem ( III.10.2 ) the formula μ ( E ) = √2f ( s ) 2 ( ds ) E establishes an isometric isomorphism between ca ( S , E , λ ) ...
Page 308
... ca ( S , E ) is weakly complete . PROOF . If { u } is a weak Cauchy sequence in ca ( S , E ) then the limit lim ( E ) exists for every E in Σ and , by II.3.27 , the sequence { u } 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 , E ) then the limit lim ( E ) exists for every E in Σ and , by II.3.27 , the sequence { u } is bounded . According to Corollary III.7.4 the ...
Page 499
... ( s ) \ v ( μ , ds ) T = x≤1 sup v ( x * ( ) x , S ) | x | ≤1 4 sup supa * ( E ) x | || 51 ΕΕΣ = 4 sup │x * ( E ) ... ca ( S , 2 , 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 ) T = x≤1 sup v ( x * ( ) x , S ) | x | ≤1 4 sup supa * ( E ) x | || 51 ΕΕΣ = 4 sup │x * ( E ) ... ca ( S , 2 , μ ) of ca ( S , E ) which consists of all u - continuous func- tions in ca ( S , E ) . By the general Radon ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ