Linear Operators: General theory |
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Page 297
... ( S , E , u ) with │x * | ≤ | 2 | . 0 If E¿ , i = 1 , 2 , . . . , n are disjoint sets in 2 with 12 ( E ) | > | λ ... Lp ( S , E , μ ) . Then for л -- the characteristic function of { E1 .... , En } , we have n U „ ƒ = Σ { μ ( E IV.8.17 ...
... ( S , E , u ) with │x * | ≤ | 2 | . 0 If E¿ , i = 1 , 2 , . . . , n are disjoint sets in 2 with 12 ( E ) | > | λ ... Lp ( S , E , μ ) . Then for л -- the characteristic function of { E1 .... , En } , we have n U „ ƒ = Σ { μ ( E IV.8.17 ...
Page 342
... Lp ( S , E , μ ) where 1 < p < ∞ , and let Σ be a family of sets of finite measure whose characteristic functions form a fundamental set in L , ( S , E , u ) . Then the sequence { f } converges to f weakly if and only if it is bounded ...
... Lp ( S , E , μ ) where 1 < p < ∞ , and let Σ be a family of sets of finite measure whose characteristic functions form a fundamental set in L , ( S , E , u ) . Then the sequence { f } converges to f weakly if and only if it is bounded ...
Page 703
... S ( x1 , ... , xx ) | ∞ 1 , and thus , by the Riesz convexity theorem , that S ( x1 , ... , k ) p≤ 1. It is clear ... Lp , then k S α ... S ( x1 , ... , x ) \ ƒ ( · ) dx1 ... dx , μ ( { s \ f ** ( s ) > ß } ) ≤ Salem | f ( s ) u ( ds ) ...
... S ( x1 , ... , xx ) | ∞ 1 , and thus , by the Riesz convexity theorem , that S ( x1 , ... , k ) p≤ 1. It is clear ... Lp , then k S α ... S ( x1 , ... , x ) \ ƒ ( · ) dx1 ... dx , μ ( { s \ f ** ( s ) > ß } ) ≤ Salem | f ( s ) u ( ds ) ...
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