Linear Operators: General theory |
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Page 95
... real function , complex function , etc. , where the adjectives real and ... valued function and ƒ a vector ( or scalar ) valued function . Thus , even ... extended real number system ( which is not a vector space ) but , since ∞ + ...
... real function , complex function , etc. , where the adjectives real and ... valued function and ƒ a vector ( or scalar ) valued function . Thus , even ... extended real number system ( which is not a vector space ) but , since ∞ + ...
Page 126
... valued , complex valued , or extended real valued additive set function defined on a field Σ of subsets of a set S. Then u is said to be countably additive if 00 μ ( UE ) = u ( E ) i = 1 i = 1 Σ whenever E , Eg , ... are disjoint sets ...
... valued , complex valued , or extended real valued additive set function defined on a field Σ of subsets of a set S. Then u is said to be countably additive if 00 μ ( UE ) = u ( E ) i = 1 i = 1 Σ whenever E , Eg , ... are disjoint sets ...
Page 137
... extended real valued set function on a field is regular . Moreover , the positive and negative variations of a bounded regular real valued additive set function are also regular . PROOF . If u is a regular additive complex or extended real ...
... extended real valued set function on a field is regular . Moreover , the positive and negative variations of a bounded regular real valued additive set function are also regular . PROOF . If u is a regular additive complex or extended real ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
Copyright | |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ