Linear Operators: General theory |
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Page 3
... function f assigns an element f ( a ) e B. If f : 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 f : 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 104
... of equivalence classes is well defined . It is customary to speak of the elements of F ( S , E , μ , 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 , E , μ , 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 μ - measurable 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 ...
... F is a μ - measurable 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 ...
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