Linear Operators: General theory |
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Page 12
... closure of A in this topology . PROOF . Statements 10 ( b ) , ( d ) show that p , Xe F , and 10 ( a ) shows that A Be F if A , B e F. It follows that any finite union of sets in F is in F. From 10 ( a ) , it follows that ACB implies AC ...
... closure of A in this topology . PROOF . Statements 10 ( b ) , ( d ) show that p , Xe F , and 10 ( a ) shows that A Be F if A , B e F. It follows that any finite union of sets in F is in F. From 10 ( a ) , it follows that ACB implies AC ...
Page 432
... closure of x ( A ) , there is an element z1 € A such that = \ x * ( ≈m ) −x ** ( x * ) \ < 1 / m , i = 1 , . . . , n . Since A is weakly sequentially compact , a subsequence of { m } will converge weakly to an element z which is ...
... closure of x ( A ) , there is an element z1 € A such that = \ x * ( ≈m ) −x ** ( x * ) \ < 1 / m , i = 1 , . . . , n . Since A is weakly sequentially compact , a subsequence of { m } will converge weakly to an element z which is ...
Page 511
... closure of co ( 4 ) . If A is compact in the strong operator topology , so is the strong operator closure of co ( 4 ) . 4 If a set AC B ( X , Y ) is sequentially compact in the weak operator topology , its weak operator closure is ...
... closure of co ( 4 ) . If A is compact in the strong operator topology , so is the strong operator closure of co ( 4 ) . 4 If a set AC B ( X , Y ) is sequentially compact in the weak operator topology , its weak operator closure is ...
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