Linear Operators: General theory |
From inside the book
Results 1-3 of 87
Page 3
... 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 C A , the symbol f ( C ) is used for the set of ...
... 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 C A , the symbol f ( C ) is used for the set 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 ...
Page 199
... function of t , is equal to the element Ss F ( s ) u ( ds ) of L „ ( T , Σ TM , λ , X ) . ' T 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 ...
... function of t , is equal to the element Ss F ( s ) u ( ds ) of L „ ( T , Σ TM , λ , X ) . ' T 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 ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
31 other sections not shown
Other editions - View all
Common terms and phrases
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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ