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 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 ...
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
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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 compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality 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 o-field 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 topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ