Linear Operators: General theory |
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Page 306
... ca ( S , E , λ ) 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 ) 2 ( ds ) establishes an isometric isomorphism between ca ( S , E , λ ) ...
... ca ( S , E , λ ) 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 ) 2 ( ds ) establishes an isometric isomorphism between ca ( S , E , λ ) ...
Page 308
... ca ( S , E ) is weakly complete . PROOF . If { n } is a weak Cauchy sequence in ca ( S , Σ ) then the limit lim μ ( E ) exists for every E in Σ 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 { n } is a weak Cauchy sequence in ca ( S , Σ ) then the limit lim μ ( E ) exists for every E in Σ and , by II.3.27 , the sequence { n } is bounded . According to Corollary III.7.4 the ...
Page 499
... ( s ) \ v ( μ , ds ) │T = sup v ( x * ( · ) x , S ) || ≤1 4 sup sup * ( E ) x | = 4 sup x * ( E ) . || 51 ΕΕΣ ΕΕΣ ... ca ( S , E , μ ) of ca ( S , Σ ) which consists of all u - continuous func- tions in ca ( S , E ) . By the general ...
... ( s ) \ v ( μ , ds ) │T = sup v ( x * ( · ) x , S ) || ≤1 4 sup sup * ( E ) x | = 4 sup x * ( E ) . || 51 ΕΕΣ ΕΕΣ ... ca ( S , E , μ ) of ca ( S , Σ ) which consists of all u - continuous func- tions in ca ( S , E ) . By the general ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite 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 theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ