Linear Operators: General theory |
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Page 161
... ba ( S , E , X ) is complete . It follows , therefore , that ba ( S , E , 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 , E , 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 , Σ ) converge weakly to an element λ e ba ( S , 2 ) if and only if there exists a non- negative u e ba ( S , 2 ) ...
... S be a compact Hausdorff space . Show that C ( S ) is weakly complete if and only if it is finite dimensional ... ba ( S , Σ ) converge weakly to an element λ e ba ( S , 2 ) if and only if there exists a non- negative u e ba ( S , 2 ) ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ