Linear Operators: General theory |
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Page 271
... shown that there exists n1 , n , no such that .. min \ ƒ ( $ n , ) — f ( $ % ) < ε , 1≤i≤r fe For which proves the quasi - uniform convergence of ƒ ( s , ) to ƒ ( s ) . Thus ( 3 ) implies ( 4 ) . We now show that ( 4 ) implies ( 5 ) ...
... shown that there exists n1 , n , no such that .. min \ ƒ ( $ n , ) — f ( $ % ) < ε , 1≤i≤r fe For which proves the quasi - uniform convergence of ƒ ( s , ) to ƒ ( s ) . Thus ( 3 ) implies ( 4 ) . We now show that ( 4 ) implies ( 5 ) ...
Page 291
... shown that L1 is weakly com- plete . Since { f } is a weak Cauchy sequence in L1 ( E ) ( by II.3.11 ) the whole argument may be made in the space L1 ( E ) . Thus , without loss of generality we may assume that ( S , E , μ ) is o ...
... shown that L1 is weakly com- plete . Since { f } is a weak Cauchy sequence in L1 ( E ) ( by II.3.11 ) the whole argument may be made in the space L1 ( E ) . Thus , without loss of generality we may assume that ( S , E , μ ) is o ...
Page 684
... shown that g is u - integra- ble . It was observed in the preceding proof that m ( q - 1e ) = m ( e ) for e in and thus the mean ergodic theorem ( 5.9 ) shows that g is in L , ( S , E , m ) . Since go g , sets of the form e = { sa < g ...
... shown that g is u - integra- ble . It was observed in the preceding proof that m ( q - 1e ) = m ( e ) for e in and thus the mean ergodic theorem ( 5.9 ) shows that g is in L , ( S , E , m ) . Since go g , sets of the form e = { sa < g ...
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