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 § ( x * ) y x * y , y € Y. Clear- ly ( x * ) ≤ x * , so that : X * → Y * . Let ŋ : 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 § ( x * ) y x * y , y € Y. Clear- ly ( x * ) ≤ x * , so that : X * → Y * . Let ŋ : y ...
Page 69
... reflexive space is weakly complete . " PROOF . If { a } is a sequence in a reflexive space X for which lim x * x , exists for each a * in X * , then , by Theorem 20 , { x } is bounded . 1300 n By Theorem 28 , there is a subsequence { a } ...
... reflexive space is weakly complete . " PROOF . If { a } is a sequence in a reflexive space X for which lim x * x , exists for each a * in X * , then , by Theorem 20 , { x } is bounded . 1300 n By Theorem 28 , there is a subsequence { a } ...
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 ...
Contents
Preliminary Concepts | 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 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ