Linear Operators: General theory |
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Page 483
... weakly compact , then TS is compact in the Y * topology of Y and thus z ( TS ) is compact and hence closed in the * topology of ** . Thus if T is weakly compact , ( i ) yields T ** ( S1 ) ≤ × ( TS ) . According to Theorem V.4.5 , S1 ...
... weakly compact , then TS is compact in the Y * topology of Y and thus z ( TS ) is compact and hence closed in the * topology of ** . Thus if T is weakly compact , ( i ) yields T ** ( S1 ) ≤ × ( 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 507
... weakly compact operator on L ( S , E , μ ) to a separable sub- set of the B - space X. Then there exists a u - essentially unique bounded measurable function x ( · ) on S to a weakly compact subset of X such that [ * ] Tf = √ ̧ x ( s ) ...
... weakly compact operator on L ( S , E , μ ) to a separable sub- set of the B - space X. Then there exists a u - essentially unique bounded measurable function x ( · ) on S to a weakly compact subset of X such that [ * ] Tf = √ ̧ x ( s ) ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ