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... function f assigns an element f ( a ) e B. If ƒ : A → B and g : B → C , then the mapping gf : A → C is defined by the equation ( gf ) ( a ) g ( f ( a ) ) for a e A. If ƒ : A → B and C CA , the symbol f ( C ) is used for the set of all ...
... function f assigns an element f ( a ) e B. If ƒ : A → B and g : B → C , then the mapping gf : A → C is defined by the equation ( gf ) ( a ) g ( f ( a ) ) for a e A. If ƒ : A → B and C CA , the symbol f ( C ) is used for the set of all ...
Page 196
... F is a u - measurable function whose values are in L2 ( T , Σr , λ ) , 1 ≤ p < . 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 ...
... F is a u - measurable function whose values are in L2 ( T , Σr , λ ) , 1 ≤ p < . 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 ...
Page 199
... function of t , is equal to the element Ss F ( s ) u ( ds ) of L2 ( T , ET , λ , X ) . PROOF . Let Σ be partitioned into a sequence { E } of disjoint sets of finite 2 - measure . For 1 ≤ p ≤ let L L ( T , ET , λ , X ) and define the ...
... function of t , is equal to the element Ss F ( s ) u ( ds ) of L2 ( T , ET , λ , X ) . PROOF . Let Σ be partitioned into a sequence { E } of disjoint sets of finite 2 - measure . For 1 ≤ p ≤ let L L ( T , ET , λ , X ) and define the ...
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