Linear Operators: General theory |
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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 X 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 X 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 , E ) . 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 , Σ ) of B ( S ) . According to Theorems 6.18 ...
... ba ( S , E ) . 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 , Σ ) of B ( S ) . According to Theorems 6.18 ...
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 , 2 ) 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 λ e ba ( S , 2 ) 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 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