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 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 which ...
... 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 which ...
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
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