FORMAL REALIZAHILITY THEORY 1 205 



whoi-e Ziip) has no pole at tcoo or — tcoo . The symmetry of Z and hnear 

 iii<lepoii(lcii('e of the terms above then implj' the symmetry of /A) , Ri 

 and Ziip). 



For any AeK, now, 



iZ{p)k, k) = ^— (/?oA-, /,■) + — ^ (/?iA-. k) + (Zi(p)A-, k). 



Applying 1(1.12 and 15.15, 



(/?oA-, A-) = (/AA-, A') > 



for all A-. Hence Ro — Ri ^ R (say) and R is semi-definite. Also, 

 (Zi(p)A-, A) appears as the residue fi{p) in 15.15 and is therefore PR. 

 Then Zi{p) is PR by 16.13. With Ro and Ri identified, (1) above is the 

 expansion given in 16.05. We have now proved all of 16.05 save the 

 reality of R. But 



p -\- OOo 



is PR, by 10.13, hence is real for real p. Therefore R is real. 

 The proof of 16.06 is similar. 



1 (').(■) To prove 16.02 we appear to digress somewhat, by first com- 

 pleting the proof of the fundamental lemma of 12.0. It was established 

 in Section 13 that the matrix [Zi{p)] describing M(p) in the chosen 

 basis is PR. The case in which it is nonsingular (i.e., m 9^ 0, cf. 12.56, 

 12.57) remains to be examined. 



16.61 If [Ziip)] is nonsingular then its inverse is PR (16.3). Then for 

 any veW2 , 



(v, k) = (v, Y{p)v) (2) 



is PR (16.12 dual). By 12.01, for any ?/eVL , the values of the function 

 F,Xp) are the values of (2) for some reV2 . Hence Fuip) is PR. This is 

 PO(A) and P7(A) for L. 

 10. ()2 To settle P5 for L in 12.0, consider peT^ and 



[i',k]dAp), [uJ]eL{p), 



where u and v are real. Then, say, 



V = Vo + Vi , 



where t'oeV^o , VieV-z . But then 



V = V = Vo -{- h 



