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Page 304
T : L , → Lg is positive and write T Z 0 when f € L , and í 2 0 imply that Tf 20. Similarly T , 2T , or T , T , means that 1 , -1 , 2 0 . It readily seen that if T Z 0 , then T maps real functions into real functions . 25 LEMMA .
T : L , → Lg is positive and write T Z 0 when f € L , and í 2 0 imply that Tf 20. Similarly T , 2T , or T , T , means that 1 , -1 , 2 0 . It readily seen that if T Z 0 , then T maps real functions into real functions . 25 LEMMA .
Page 305
We now show that T , is additive on positive elements . Let f ; e L , ( S , E , u ) , ti 20 , j = 1 , 2 , and let f1 = { i = 1811 , 12 Ei = 1 & zi be decompositions of 11 and 12 into positive functions .
We now show that T , is additive on positive elements . Let f ; e L , ( S , E , u ) , ti 20 , j = 1 , 2 , and let f1 = { i = 1811 , 12 Ei = 1 & zi be decompositions of 11 and 12 into positive functions .
Page 714
( a ) there exists a positive integer N such that TNx = x , for x e Cp ; ( b ) Tnx → 0 exponentially fast , for x € Co. Moreover , the subspaces Cp and C , are uniquely defined by properties ( a ) and ( b ) , respectively . PROOF .
( a ) there exists a positive integer N such that TNx = x , for x e Cp ; ( b ) Tnx → 0 exponentially fast , for x € Co. Moreover , the subspaces Cp and C , are uniquely defined by properties ( a ) and ( b ) , respectively . PROOF .
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Contents
Preliminary Concepts | 1 |
The VitaliHahnSaks Theorem and Spaces of Measures | 7 |
B Topological Preliminaries | 10 |
Copyright | |
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