Linear Operators: General theory |
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Page 306
... ca ( S , Σ , λ ) consisting of all 2 - continuous functions in ca ( S , 2 ) . According to the Radon - Nikodým theorem ( III.10.2 ) the formula μ ( E ) = √ † ( s ) ż ( ds ) E establishes an isometric isomorphism between ca ( S , E , λ ) ...
... ca ( S , Σ , λ ) consisting of all 2 - continuous functions in ca ( S , 2 ) . According to the Radon - Nikodým theorem ( III.10.2 ) the formula μ ( E ) = √ † ( s ) ż ( ds ) E 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 , E ) then the limit lim ( E ) exists for every E in 2 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 , E ) then the limit lim ( E ) exists for every E in 2 and , by II.3.27 , the sequence { n } is bounded . According to Corollary III.7.4 the ...
Page 499
... ( s ) | v ( μ , ds ) | x | ≤1 │T - ≤1 sup v ( x * ( ) x , S ) x ≤1 4 sup supa * ( E ) x = 4 sup x * ( E ) . || 51 ... ca ( S , Σ , μ ) of ca ( S , E ) which consists of all μ - continuous func- tions in ca ( S , E ) . By the general ...
... ( s ) | v ( μ , ds ) | x | ≤1 │T - ≤1 sup v ( x * ( ) x , S ) x ≤1 4 sup supa * ( E ) x = 4 sup x * ( E ) . || 51 ... ca ( S , Σ , μ ) of ca ( S , E ) which consists of all μ - continuous func- tions in ca ( S , E ) . By the general ...
Contents
Preliminary Concepts | 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 ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f 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 S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ