Linear Operators: General theory |
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Page 95
... contains at most one of the improper values + and . A positive set function is a real valued or extended real valued set function which has no negative values . 2 DEFINITION . A set function u defined on a 95 Integration and Set Functions.
... contains at most one of the improper values + and . A positive set function is a real valued or extended real valued set function which has no negative values . 2 DEFINITION . A set function u defined on a 95 Integration and Set Functions.
Page 119
... function f ( 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 ¢ N , g ( s ) = 0 if se N , is u ...
... function f ( 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 ¢ N , g ( s ) = 0 if se N , is u ...
Page 196
... valued integrals . In the application of the theory of vector valued integrals to concrete problems such as the ... function whose values are in L2 ( T , Σr , λ ) , 1 ≤ p < . For each s in S , F ( s ) is an equivalence class of functions , ...
... valued integrals . In the application of the theory of vector valued integrals to concrete problems such as the ... function whose values are in L2 ( T , Σr , λ ) , 1 ≤ p < . For each s in S , F ( s ) is an equivalence class of functions , ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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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