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 Tx is finite , then for an arbitrary x 0 , | x | ≤1 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 Tx is finite , then for an arbitrary x 0 , | x | ≤1 Tx = x T ≤Mx ...
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 { a } , so that , by Lemma II.1.5 , 2 , 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 { a } , so that , by Lemma II.1.5 , 2 , is sep- arable . By Theorem 5.1 and ...
Page 454
... implies N ( p ) = q . We note that N ( C ) CK , while pe K implies that N ( p ) = p . Now , if T K → K is continuous , TN : C → K is continuous and by the preceding lemma has a fixed point . This fixed point is in K ; it is therefore ...
... implies N ( p ) = q . We note that N ( C ) CK , while pe K implies that N ( p ) = p . Now , if T K → K is continuous , TN : C → K is continuous and by the preceding lemma has a fixed point . This fixed point is in K ; it is therefore ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz 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