Linear Operators: General theory |
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Page 69
... preceding lemma . Q.E.D. 29 COROLLARY . A reflexive space is weakly complete . " PROOF . If { a } is a sequence in a reflexive space X for which lim x * x , exists for each a * in X * , then , by Theorem 20 , { x } is bounded . 1300 n By ...
... preceding lemma . Q.E.D. 29 COROLLARY . A reflexive space is weakly complete . " PROOF . If { a } is a sequence in a reflexive space X for which lim x * x , exists for each a * in X * , then , by Theorem 20 , { x } is bounded . 1300 n By ...
Page 277
... preceding theorem has no simple or familiar representation . 23 DEFINITION . A linear functional * on an algebra A will be called multiplicative if x * ( fg ) = ( x * f ) ( x * g ) for every ƒ and g in A. 24 LEMMA . Let A be a closed ...
... preceding theorem has no simple or familiar representation . 23 DEFINITION . A linear functional * on an algebra A will be called multiplicative if x * ( fg ) = ( x * f ) ( x * g ) for every ƒ and g in A. 24 LEMMA . Let A be a closed ...
Page 681
... preceding theorem is not true if p = 1 , but there is a k - para- meter analogue of Lemma 5. This will be found at ... LEMMA . Let ( S , E , μ ) be a finite positive measure space . A bounded sequence { u } in the B - space of μ ...
... preceding theorem is not true if p = 1 , but there is a k - para- meter analogue of Lemma 5. This will be found at ... LEMMA . Let ( S , E , μ ) be a finite positive measure space . A bounded sequence { u } in the B - space of μ ...
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