FORMAL REALIZABILITY THEORY — I 271 



Therefore ] m(p), the space of currents admitted by M at frequency p, 

 coincides with the fixed J.u at each peFi . 

 Clearly J a/ is a real linear manifold. 



18.32 Consider now U.uo(p): if [u, 0]eM(p), then there is a t; such that 

 u = C*v and 



[v, CO] = [v, 0]eL(p). 



Hence veWwip) = Vlo for each peYt . Therefore, for each pcTi, , 



VUp) CC*V.o. (2) 



Now suppose, conversely, that peT^ and t'eVio = 'Vlo(p)- Then 

 [v, 0]eL{p). Now = CO, so [v, CO]eL{p). Hence [C*v, 0]eM(p), so 

 C*velJ Mo(p). This proves the inequality opposite to that of (2), so for peTi, 



U.vo(p) = C*V,o = Umo, (3) 



a fixed space. 



18.33 Now consider (U.^o)". If ie(U,/o)", then 



(w,i)2 = 



for everj^ weXJ^/o . That is, by (3), 



iC*v,j), = (v,Cj), = 



for every vcVlo . Therefore Cjiiyuof = Kz, , and je] m by 18.31. That 

 is, we have proved 



and, combining 18.31 with this and (3), 



]^f{p) = J.U 2 (U.vo(?>))" = (U.v,o)". (4) 



This is the weak form P3'(I) of 12.0 for M. It is as far as we can go 

 with P3 at the moment. 



18.34 Consider P4. If for peTL we have 



[u, JhMip) 



then [v, Cj]eL(p) and u = C*v. But then [v, Cj]eL(p) and ii = C*v, by 

 18.12. Then however 



[uJ]eM(p) 

 by definition of M. This is P4. 



