Linear Operators: General theory |
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Page 358
... strongly in L1 if and only if Sn → I strongly in C. n 5 Show that S → I strongly in L , if and only if S → I strongly in La , ( 1 / p + 1 / q = 1 ) . 6 Show that ( Sf ) ( x ) converges to f ( x ) uniformly on [ 0 , 27 ] for all fe C ...
... strongly in L1 if and only if Sn → I strongly in C. n 5 Show that S → I strongly in L , if and only if S → I strongly in La , ( 1 / p + 1 / q = 1 ) . 6 Show that ( Sf ) ( x ) converges to f ( x ) uniformly on [ 0 , 27 ] for all fe C ...
Page 685
... strongly measurable if , for each a in X , the function T ( · ) æ is measurable , with respect to Lebesgue meas- ure , on the infinite interval 0 ≤t . It was observed in Lemma 1.3 that a strongly measurable semi - group is strongly ...
... strongly measurable if , for each a in X , the function T ( · ) æ is measurable , with respect to Lebesgue meas- ure , on the infinite interval 0 ≤t . It was observed in Lemma 1.3 that a strongly measurable semi - group is strongly ...
Page 689
... strongly convergent if they are bounded and if T ( n ) / n converges to zero strongly . PROOF . This follows from Theorem 1 and Theorem II.3.28 . Q.E.D. 4 COROLLARY . Let ( S , E , u ) be a finite positive measure space and { T ( t ) ...
... strongly convergent if they are bounded and if T ( n ) / n converges to zero strongly . PROOF . This follows from Theorem 1 and Theorem II.3.28 . Q.E.D. 4 COROLLARY . Let ( S , E , u ) be a finite positive measure space and { T ( t ) ...
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