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 4 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 4 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 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 = S ...
... 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 = S ...
Page 485
... compact if and only if its adjoint is weakly compact . PROOF . Let T be weakly compact . Since the closed unit sphere S * of Y * is Y - compact ( V.4.2 ) , it follows from Lemma 7 and Lemma I.5.7 that T * S * is compact in the X ...
... compact if and only if its adjoint is weakly compact . PROOF . Let T be weakly compact . Since the closed unit sphere S * of Y * is Y - compact ( V.4.2 ) , it follows from Lemma 7 and Lemma I.5.7 that T * S * is compact in the X ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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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