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 | ≤1 x │Tx | = | x ...
... 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 x │Tx | = | x ...
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 ( ii ) . The other implications we desire are decidedly non - trivial ; we will complete the proof by showing first that ( ii ) implies ( i ) , and then that ( i ) implies ( iii ) . arable . By Theorem 5.1 - Proof that condition ...
... implies ( ii ) . The other implications we desire are decidedly non - trivial ; we will complete the proof by showing first that ( ii ) implies ( i ) , and then that ( i ) implies ( iii ) . arable . By Theorem 5.1 - Proof that condition ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ