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 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