Linear Operators: General theory |
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Page 104
... of equivalence classes is well defined . It is customary to speak of the elements of F ( S , 2 , μ , X ) as if they were functions rather than sets of equivalent functions and this we shall ordinarily do . Thus , we shall write f instead of ...
... of equivalence classes is well defined . It is customary to speak of the elements of F ( S , 2 , μ , X ) as if they were functions rather than sets of equivalent functions and this we shall ordinarily do . Thus , we shall write f instead of ...
Page 196
... F is a μ - measurable function whose values are in L ( T , ΣT , 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 μ - measurable function whose values are in L ( T , ΣT , 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 fs F ( s ) u ( ds ) of L „ ( T , Σr , λ , X ) . T PROOF . Let Σ be partitioned into a sequence { E } of disjoint sets of finite 2 - measure . For 1 ≤ p ≤ let L , = L ( T , ET , 2. X ) and define ...
... function of t , is equal to the element fs F ( s ) u ( ds ) of L „ ( T , Σr , λ , X ) . T PROOF . Let Σ be partitioned into a sequence { E } of disjoint sets of finite 2 - measure . For 1 ≤ p ≤ let L , = L ( T , ET , 2. X ) and define ...
Contents
Preliminary Concepts | 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 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ