Linear Operators: General theory |
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Page 69
... weakly to y . Hence bounded sets are weakly sequentially compact . The converse follows from the preceding lemma . Q.E.D. 29 COROLLARY . A reflexive space is weakly complete . " PROOF . If { a } is a sequence in a reflexive space X for ...
... weakly to y . Hence bounded sets are weakly sequentially compact . The converse follows from the preceding lemma . Q.E.D. 29 COROLLARY . A reflexive space is weakly complete . " PROOF . If { a } is a sequence in a reflexive space X for ...
Page 507
... weakly compact subset of X , then equation [ * ] defines a weakly compact mapping of L1 ( S , E , μ ) into X with separable range . PROOF . Let U be the closed unit sphere of L1 ( S , Σ , μ ) . If T is weakly compact and has a separable ...
... weakly compact subset of X , then equation [ * ] defines a weakly compact mapping of L1 ( S , E , μ ) into X with separable range . PROOF . Let U be the closed unit sphere of L1 ( S , Σ , μ ) . If T is weakly compact and has a separable ...
Page 540
... Weakly compact operators from C [ 0 , 1 ] to X were treated by Sirvint [ 3 ] . A very incisive discussion of weakly compact operators with domain C ( S ) was given by Grothendieck [ 4 ] who proved Theorems 7.4-7.6 by other means ...
... Weakly compact operators from C [ 0 , 1 ] to X were treated by Sirvint [ 3 ] . A very incisive discussion of weakly compact operators with domain C ( S ) was given by Grothendieck [ 4 ] who proved Theorems 7.4-7.6 by other means ...
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