Linear Operators: General theory |
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Page 358
... strongly in L1 if and only if Sn → I strongly in C. Sn 5 Show that S , → I strongly in L , if and only if S → I strongly in La ( 1 / p + 1 / q = 1 ) . 6 Show that ( Sf ) ( x ) converges to f ( x ) uniformly on [ 0 , 27 ] for all fe C ...
... strongly in L1 if and only if Sn → I strongly in C. Sn 5 Show that S , → I strongly in L , if and only if S → I strongly in La ( 1 / p + 1 / q = 1 ) . 6 Show that ( Sf ) ( x ) converges to f ( x ) uniformly on [ 0 , 27 ] for all fe C ...
Page 472
... strongly differentiable at the point x . Banach [ 1 ; p . 168 ] showed that the norm in C [ 0 , 1 ] is strongly differentiable at xo e C [ 0 , 1 ] if and only if the function x achieves its maximum at exactly one point . Mazur [ 1 ; p ...
... strongly differentiable at the point x . Banach [ 1 ; p . 168 ] showed that the norm in C [ 0 , 1 ] is strongly differentiable at xo e C [ 0 , 1 ] if and only if the function x achieves its maximum at exactly one point . Mazur [ 1 ; p ...
Page 685
... strongly measurable if , for each a in X , the function T ( ) is measurable , with respect to Lebesgue meas- ure , on the infinite interval 0 ≤t . It was observed in Lemma 1.3 that a strongly measurable semi - group is strongly ...
... strongly measurable if , for each a in X , the function T ( ) is measurable , with respect to Lebesgue meas- ure , on the infinite interval 0 ≤t . It was observed in Lemma 1.3 that a strongly measurable semi - group is strongly ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian 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