Linear Operators: General theory |
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Page 17
... compact if its closure is compact in its relative topology . It should be noted that a subset ACX is compact in its relative topology if and only if every covering of A by open sets in X contains a finite subcovering . A well - known ...
... compact if its closure is compact in its relative topology . It should be noted that a subset ACX is compact in its relative topology if and only if every covering of A by open sets in X contains a finite subcovering . A well - known ...
Page 483
... compact , then TS is compact in the Y * topology of Y and thus ( TS ) is compact and hence closed in the Y * topology of Y ** . Thus if T is weakly compact , ( i ) yields T ** ( S1 ) ≤ × ( TS ) . According to Theorem V.4.5 , S1 = S ...
... compact , then TS is compact in the Y * topology of Y and thus ( TS ) is compact and hence closed in the Y * topology of Y ** . Thus if T is weakly compact , ( i ) yields T ** ( S1 ) ≤ × ( TS ) . According to Theorem V.4.5 , S1 = S ...
Page 494
... 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 Cauchy sequences into strongly convergent sequences ...
... 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 Cauchy sequences into strongly convergent sequences ...
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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian 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