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Page 161
... ba ( S , Σ , X ) is complete . It follows , therefore , that ba ( S , Σ , X ) is a B - space . If is the set of real or complex numbers , then according to Lemma 1.5 , sup │μ ( E ) | ≤ v ( μ , S ) ≤ 4 sup | μ ( E ) | . ΕΕΣ ΕΕΣ This ...
... ba ( S , Σ , X ) is complete . It follows , therefore , that ba ( S , Σ , X ) is a B - space . If is the set of real or complex numbers , then according to Lemma 1.5 , sup │μ ( E ) | ≤ v ( μ , S ) ≤ 4 sup | μ ( E ) | . ΕΕΣ ΕΕΣ This ...
Page 311
... ba ( S , Σ ) . 9 THEOREM . The space ba ( S , E ) is weakly complete . If S is a topological space , the rba ( S ) is also weakly complete . PROOF . Consider the closed subspace B ( S , E ) of B ( S ) . According to Theorems 6.18 and ...
... ba ( S , Σ ) . 9 THEOREM . The space ba ( S , E ) is weakly complete . If S is a topological space , the rba ( S ) is also weakly complete . PROOF . Consider the closed subspace B ( S , E ) of B ( S ) . According to Theorems 6.18 and ...
Page 340
... S be a compact Hausdorff space . Show that C ( S ) is weakly complete if and only if it is finite dimensional ... ba ( S , E ) converge weakly to an element λ e ba ( S , Σ ) if and only if there exists a non- negative u e ba ( S , E ) ...
... S be a compact Hausdorff space . Show that C ( S ) is weakly complete if and only if it is finite dimensional ... ba ( S , E ) converge weakly to an element λ e ba ( S , Σ ) if and only if there exists a non- negative u e ba ( S , E ) ...
Contents
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