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Page 125
... ( s ) | ≤g ( s ) | almost everywhere for each a . Then fa converges to f in u - measure if and only if f is in L ... Lp ( S , E , μ , X ) are metric . Let fa → f in measure and suppose that ƒ does not converge to fin L ,. Then there is ...
... ( s ) | ≤g ( s ) | almost everywhere for each a . Then fa converges to f in u - measure if and only if f is in L ... Lp ( S , E , μ , X ) are metric . Let fa → f in measure and suppose that ƒ does not converge to fin L ,. Then there is ...
Page 288
... ( S ) = - Ent ( s ) u - almost everywhere on E , the limit g ( s ) n + 1 = lim , g ( s ) exists μ - almost everywhere ... Lp . It has been shown that g , ≤ x * . On the other hand it follows from Hölder's inequality that * ≤ g ,. Thus x * → ...
... ( S ) = - Ent ( s ) u - almost everywhere on E , the limit g ( s ) n + 1 = lim , g ( s ) exists μ - almost everywhere ... Lp . It has been shown that g , ≤ x * . On the other hand it follows from Hölder's inequality that * ≤ g ,. Thus x * → ...
Page 297
... ( s ) λ ( ds ) | ≤ | ƒ || 2 | . Thus , equation [ * ] does define an element ** € L * ( S , Σ , u ) with | x ... Lp ( S , E , u ) . Then for = { E ,, ... , En , we have л n U2t = Σ { μ ( E , ) IV.8.17 297 SPACES L , ( S , Σ , μ )
... ( s ) λ ( ds ) | ≤ | ƒ || 2 | . Thus , equation [ * ] does define an element ** € L * ( S , Σ , u ) with | x ... Lp ( S , E , u ) . Then for = { E ,, ... , En , we have л n U2t = Σ { μ ( E , ) IV.8.17 297 SPACES L , ( S , Σ , μ )
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz 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