Linear Operators: General theory |
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Page 67
... reflexive B - space is reflexive . PROOF . Let be a closed linear manifold in the reflexive space X. Let § : x * → y * be defined by the equation ( * ) y = x * y , y € Y. Clear- ye ly ( x * x * , so that : X * → Y * . Let n : y ...
... reflexive B - space is reflexive . PROOF . Let be a closed linear manifold in the reflexive space X. Let § : x * → y * be defined by the equation ( * ) y = x * y , y € Y. Clear- ye ly ( x * x * , so that : X * → Y * . Let n : y ...
Page 88
... reflexive . Examples of reflexive B - spaces are given in Chapter IV , and a necessary and sufficient condition for reflexivity is given in Theorem V.4.7 . It is a consequence of Theorem V.6.1 that a necessary and sufficient condition ...
... reflexive . Examples of reflexive B - spaces are given in Chapter IV , and a necessary and sufficient condition for reflexivity is given in Theorem V.4.7 . It is a consequence of Theorem V.6.1 that a necessary and sufficient condition ...
Page 425
... reflexive spaces . 7 THEOREM . A B - space is reflexive if and only if its closed unit sphere is compact in the weak topology . PROOF . Let X be a reflexive B - space , and let z be the natural embedding of X onto X ** . Then x and x ...
... reflexive spaces . 7 THEOREM . A B - space is reflexive if and only if its closed unit sphere is compact in the weak topology . PROOF . Let X be a reflexive B - space , and let z be the natural embedding of X onto X ** . Then x and x ...
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