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Page 279
... ( t ) ) = ( Hf ) ( t ) is continuous in t , the set = U = { t | f ( h ( t ) ) = 0 } is a neighborhood of to . If t is in U then f ( h ( t ) ) 0. This shows that h ( t ) is in N. Thus h ( U ) CN and h is continuous . If H is an isomorphism ...
... ( t ) ) = ( Hf ) ( t ) is continuous in t , the set = U = { t | f ( h ( t ) ) = 0 } is a neighborhood of to . If t is in U then f ( h ( t ) ) 0. This shows that h ( t ) is in N. Thus h ( U ) CN and h is continuous . If H is an isomorphism ...
Page 579
... T are not one - to - one . Then , by Corollary 2 , only a finite number of the maps 2λI * -T * are one - to - one . However , T * is compact ( by VI.5.2 ) , and Lemma 4 , applied to T * , yields a contradiction . This proves that every ...
... T are not one - to - one . Then , by Corollary 2 , only a finite number of the maps 2λI * -T * are one - to - one . However , T * is compact ( by VI.5.2 ) , and Lemma 4 , applied to T * , yields a contradiction . This proves that every ...
Page 631
... for t≥ 0 . We will build up the proof of Theorem 19 through a series of lemmas . Throughout the remainder of this ... in t for t > 0 and each xe X. If wo = lim log T ( t ) | t , then lim sup log | PT ( t ) | two ; 047 t - ∞ ( d ) if R ...
... for t≥ 0 . We will build up the proof of Theorem 19 through a series of lemmas . Throughout the remainder of this ... in t for t > 0 and each xe X. If wo = lim log T ( t ) | t , then lim sup log | PT ( t ) | two ; 047 t - ∞ ( d ) if R ...
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