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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 x Tx = xT ≤ M ...
... 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 x Tx = xT ≤ M ...
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 ...
... 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 ...
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 ...
... 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 ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ