Linear Operators: General theory |
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Page 12
... closure of A in this topology . PROOF . Statements 10 ( b ) , ( d ) show that 4 , X € F , and 10 ( a ) shows that A Be F if A , Be F. It follows that any finite union of sets in F is in F. From 10 ( a ) , it follows that AC B implies ĀC ...
... closure of A in this topology . PROOF . Statements 10 ( b ) , ( d ) show that 4 , X € F , and 10 ( a ) shows that A Be F if A , Be F. It follows that any finite union of sets in F is in F. From 10 ( a ) , it follows that AC B implies ĀC ...
Page 462
... closure for subspaces of a normed linear space ( a special case of Theorem 3.13 ) appears in Banach [ 1 ; pp . 58 ... closure which we have studied , various special definitions of closure for subspaces or convex subsets of a linear ...
... closure for subspaces of a normed linear space ( a special case of Theorem 3.13 ) appears in Banach [ 1 ; pp . 58 ... closure which we have studied , various special definitions of closure for subspaces or convex subsets of a linear ...
Page 511
... closure of co ( A ) . 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 ( A ) . 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
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 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 disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ