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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 , Tx = | x | T │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 , Tx = | x | T │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 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