Linear Operators: General theory |
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Page 95
... function u . In some cases that will be encountered the values of μ are not scalars , but customarily where integration is used in this text μ is a scalar valued function and ƒ a vector ( or scalar ) valued function . Thus , even if the ...
... function u . In some cases that will be encountered the values of μ are not scalars , but customarily where integration is used in this text μ is a scalar valued function and ƒ a vector ( or scalar ) valued function . Thus , even if the ...
Page 119
... function ƒ ( vector or extended real - valued ) which is defined only on the complement of a u - null set NCS . Then we say that ƒ is u - measurable if the function g defined by g ( s ) = f ( s ) if s 4 N , g ( s ) = 0 if s e N , is μ ...
... function ƒ ( vector or extended real - valued ) which is defined only on the complement of a u - null set NCS . Then we say that ƒ is u - measurable if the function g defined by g ( s ) = f ( s ) if s 4 N , g ( s ) = 0 if s e N , is μ ...
Page 196
... function whose values are in L „ ( T , Σr , λ ) , 1 ≤ p < ∞o . For each s in S , F ( s ) is an equivalence class of functions , any pair of whose members coincide 2 - almost every- where . If for each s we select a particular function ...
... function whose values are in L „ ( T , Σr , λ ) , 1 ≤ p < ∞o . For each s in S , F ( s ) is an equivalence class of functions , any pair of whose members coincide 2 - almost every- where . If for each s we select a particular function ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ