Linear Operators: General theory |
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Page 97
... additive set function u is important because it dominates u in the sense that v ( u , E ) ≥ │μ ( E ) | for E e 2 ; the reader should test his com- prehension of Definition 4 below by proving that v ( u ) is the smallest of the non ...
... additive set function u is important because it dominates u in the sense that v ( u , E ) ≥ │μ ( E ) | for E e 2 ; the reader should test his com- prehension of Definition 4 below by proving that v ( u ) is the smallest of the non ...
Page 126
... additive set func- tion defined on a σ - field of subsets of a set . In this case the results of the preceding sections can be considerably extended . 1 DEFINITION . Let u be a vector valued , complex valued , or extended real valued ...
... additive set func- tion defined on a σ - field of subsets of a set . In this case the results of the preceding sections can be considerably extended . 1 DEFINITION . Let u be a vector valued , complex valued , or extended real valued ...
Page 233
... additive integrals . In Section IV.10 we will pre- sent a Lebesgue type theory of integration of a scalar function with respect to a countably additive vector valued measure . Instances in which both the function and the measure are ...
... additive integrals . In Section IV.10 we will pre- sent a Lebesgue type theory of integration of a scalar function with respect to a countably additive vector valued measure . Instances in which both the function and the measure are ...
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