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Page 306
... ca ( S , E , λ ) consisting of all λ - continuous functions in ca ( S , E ) . According to the Radon - Nikodým theorem ( III.10.2 ) the formula μ ( E ) = √¿f ( s ) 2 ( ds ) E establishes an isometric isomorphism between ca ( S , E , λ ) ...
... ca ( S , E , λ ) consisting of all λ - continuous functions in ca ( S , E ) . According to the Radon - Nikodým theorem ( III.10.2 ) the formula μ ( E ) = √¿f ( s ) 2 ( 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 u , ( E ) exists for every E in 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 { n } is a weak Cauchy sequence in ca ( S , E ) then the limit lim u , ( E ) exists for every E in and , by II.3.27 , the sequence { u } is bounded . According to Corollary III.7.4 the ...
Page 499
... ( s ) \ v ( u , ds ) | x | ≤1 | x | ≤1 Tsup v ( x * ( · ) x , S ) | x | ≤1 4 sup sup * ( E ) x | || 51 ΕΕΣ = 4 sup ... ca ( S , 2 , μ ) of ca ( S , E ) which consists of all μ - continuous func- tions in ca ( S , E ) . By the general ...
... ( s ) \ v ( u , ds ) | x | ≤1 | x | ≤1 Tsup v ( x * ( · ) x , S ) | x | ≤1 4 sup sup * ( E ) x | || 51 ΕΕΣ = 4 sup ... ca ( S , 2 , μ ) of ca ( S , E ) which consists of all μ - continuous func- tions in ca ( S , E ) . By the general ...
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A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ