Linear Operators: General theory |
From inside the book
Results 1-3 of 33
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
... extended . DEFINITION . Let u be a vector 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 ) = Σμ ( Ε . ) i = 1 i ...
... extended . DEFINITION . Let u be a vector 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 ) = Σμ ( Ε . ) i = 1 i ...
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 μ 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 μ is a regular additive complex or extended real ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
31 other sections not shown
Other editions - View all
Common terms and phrases
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