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 - Nikodım 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 - Nikodım 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 ( E ) exists for every E in and , by II.3.27 , the sequence { n } is bounded . According to Corollary III.7.4 the countable ...
... ca ( S , E ) is weakly complete . PROOF . If { n } is a weak Cauchy sequence in ca ( S , Σ ) then the limit lim ( E ) exists for every E in and , by II.3.27 , the sequence { n } is bounded . According to Corollary III.7.4 the countable ...
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 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