Linear Operators: General theory |
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Page 483
... weakly compact , then TS is compact in the Y * topology of Y and thus x ( TS ) is compact and hence closed in the Y * topology of Y ** . Thus if T is weakly compact , ( i ) yields T ** ( S1 ) C ( TS ) . According to Theorem V.4.5 , S1 ...
... weakly compact , then TS is compact in the Y * topology of Y and thus x ( TS ) is compact and hence closed in the Y * topology of Y ** . Thus if T is weakly compact , ( i ) yields T ** ( S1 ) C ( TS ) . According to Theorem V.4.5 , S1 ...
Page 494
... weakly compact set of rca ( 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 rca ( 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 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
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 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ