Linear Operators: General theory |
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Page 494
... weakly compact set of rea ( S ) , and therefore T * is a weakly compact operator . By Theorem 4.8 this implies that T is a weakly compact operator . Q.E.D. 4 THEOREM . If T is a weakly compact operator from C ( S ) to X , then T sends weak ...
... weakly compact set of rea ( S ) , and therefore T * is a weakly compact operator . By Theorem 4.8 this implies that T is a weakly compact operator . Q.E.D. 4 THEOREM . If T is a weakly compact operator from C ( S ) to X , then T sends weak ...
Page 507
... weakly compact operator on L1 ( S , Σ , μ ) to a separable sub- set of the B - space X. Then there exists a μ - essentially unique bounded measurable function x ( · ) on S to a weakly compact subset of X such that [ * ] T † = √ ̧ x ( s ) ...
... weakly compact operator on L1 ( S , Σ , μ ) to a separable sub- set of the B - space X. Then there exists a μ - essentially unique bounded measurable function x ( · ) on S to a weakly compact subset of X such that [ * ] T † = √ ̧ x ( s ) ...
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
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ