Linear Operators: General theory |
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Page 161
... ba ( S , E , 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 , E , 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 2 e ba ( S , E ) if and only if there exists a non- negative μ 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 2 e ba ( S , E ) if and only if there exists a non- negative μ e ba ( S , E ) ...
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 converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ