Linear Operators: General theory |
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Page 60
... implies ( iv ) . Statement ( iv ) clearly implies the continuity of T at 0 ; so ( iv ) implies ( ii ) . This ( i ) , ( ii ) , and ( iv ) are equiv- alent . If M = sup Ta is finite , then for an arbitrary x 0 , x | Tx | = | x || T ≤Mx ...
... implies ( iv ) . Statement ( iv ) clearly implies the continuity of T at 0 ; so ( iv ) implies ( ii ) . This ( i ) , ( ii ) , and ( iv ) are equiv- alent . If M = sup Ta is finite , then for an arbitrary x 0 , x | Tx | = | x || T ≤Mx ...
Page 280
... implies ( 2 ) can be proved in a manner similar to that used in Theorem 14 to show that condition ( 3 ) of that theorem implies ( 4 ) . From Corollary 19 it follows that S may be embedded as a dense subset of a compact Hausdorff space ...
... implies ( 2 ) can be proved in a manner similar to that used in Theorem 14 to show that condition ( 3 ) of that theorem implies ( 4 ) . From Corollary 19 it follows that S may be embedded as a dense subset of a compact Hausdorff space ...
Page 430
... implies ( i ) , and then that ( i ) implies ( iii ) . = Proof that condition ( ii ) implies ( i ) . Let { x } be an arbitrary se- quence in A and let X。 sp { x } , so that , by Lemma II.1.5 , X , is sep- arable . By Theorem 5.1 and ...
... implies ( i ) , and then that ( i ) implies ( iii ) . = Proof that condition ( ii ) implies ( i ) . Let { x } be an arbitrary se- quence in A and let X。 sp { x } , so that , by Lemma II.1.5 , X , is sep- arable . By Theorem 5.1 and ...
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