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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 | ≤1 Tx = xT HT ...
... 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 | ≤1 Tx = xT HT ...
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 454
... implies N ( p ) = q . We note that N ( C ) CK , while pe K implies that N ( p ) = p . Now , if TK → 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 a ...
... implies N ( p ) = q . We note that N ( C ) CK , while pe K implies that N ( p ) = p . Now , if TK → 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 a ...
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 converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f 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 S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ