Linear Operators: General theory |
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Page 166
... Lp ( u ) = L , ( S , E , u , X ) . By the argument presented in the proof of Theorem 2.22 we may and shall assume that | ƒ „ ( s ) | ≤ 2 | f ( s ) | for all s in S. By the dominated conver- gence theorem , ƒn → ƒ in L „ ( μ ) , and ...
... Lp ( u ) = L , ( S , E , u , X ) . By the argument presented in the proof of Theorem 2.22 we may and shall assume that | ƒ „ ( s ) | ≤ 2 | f ( s ) | for all s in S. By the dominated conver- gence theorem , ƒn → ƒ in L „ ( μ ) , and ...
Page 297
... ( s ) λ ( ds ) | ≤ | f || 2 | . Thus , equation [ * ] does define an element ** € L * ( S , E , μ ) with │x ... Lp ( S , E , μ ) . Then for л = { E1 , ... , E } , we have == n U2 † = Σ { μ ( E , IV.8.17 297 SPACES L ( S , E , μ )
... ( s ) λ ( ds ) | ≤ | f || 2 | . Thus , equation [ * ] does define an element ** € L * ( S , E , μ ) with │x ... Lp ( S , E , μ ) . Then for л = { E1 , ... , E } , we have == n U2 † = Σ { μ ( E , IV.8.17 297 SPACES L ( S , E , μ )
Page 342
... Lp ( S , E , μ ) where 1 < p < ∞ , and let Zo be a family of sets of finite measure whose characteristic functions form a fundamental set in L , ( S , E , μ ) . Then the sequence { f } converges to f weakly if and only if it is bounded ...
... Lp ( S , E , μ ) where 1 < p < ∞ , and let Zo be a family of sets of finite measure whose characteristic functions form a fundamental set in L , ( S , E , μ ) . Then the sequence { f } converges to f weakly if and only if it is bounded ...
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