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 x € C [ 0 , 1 ] if and only if the function 。 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 x € C [ 0 , 1 ] if and only if the function 。 achieves its maximum at exactly one point . Mazur [ 1 ; p ...
Page 685
... strongly measurable if , for each a in X , the function T ( · ) x 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 ( · ) x 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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ