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Page 3
... function f assigns an element f ( a ) e B. If f : A → B and g : BC , then the mapping gf : AC is defined by the equation ( gf ) ( a ) g ( f ( a ) ) for a e A. If ƒ : A → B and C C A , the symbol f ( C ) is used for the set of all elements ...
... function f assigns an element f ( a ) e B. If f : A → B and g : BC , then the mapping gf : AC is defined by the equation ( gf ) ( a ) g ( f ( a ) ) for a e A. If ƒ : A → B and C C A , the symbol f ( C ) is used for the set of all elements ...
Page 104
... of equivalence classes is well defined . It is customary to speak of the elements of F ( S , Z , u , X ) as if they were functions rather than sets of equivalent functions and this we shall ordinarily do . Thus , we shall write ƒ instead of ...
... of equivalence classes is well defined . It is customary to speak of the elements of F ( S , Z , u , X ) as if they were functions rather than sets of equivalent functions and this we shall ordinarily do . Thus , we shall write ƒ instead of ...
Page 196
... F is a u - measurable function whose values are in L2 ( T , ET , 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 , ET , 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 ...
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 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