250 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



then the value of {v, k) is determined by k. This makes it possible to 

 state P6(I) and P7(I) for L (we take P6(I) to include the hypothesis 

 that Tl has a finite complement). 



12.11 If keKi^ , then Jk{p) is PR. 

 Proof: if k is real then 



JM = (v, k) = (v, k), (2) 



where, of course, [v, k]€L{p). Then however [v, k]eL{p), by P4. Hence 

 by 12.1, (2) gives us 



Jk{p) = J kip). 



From this and P6(I), P7(I) we conclude that Jk{p) is PR for any real 

 fceKx, . 



Now, given any keKi, , we have keK-L by 12.01. Then 



k = ki -\- ik-2 



where ki and k-i are real and in K^ , since Kl is a linear manifold (see 

 10.42). Let 



[Vr , kr]eL(p), 



r = 1, 2. Then we have (P2) 



[vi + iv2 , k]eL{p). 

 Then 



'^kip) = {vi , ki) + (vo , k-i) + i{vi , k'2) — i{v2 , h). 



Now by P5(I), {vi , k-i) = {ih , ki). Hence 



J,(p) = (v, , /m) + (v, , b^ (3) 



for any peTL . Since each summand in (3) is a PR function, it follows 

 that Jk(p) is PR for any AeK. 



12.12 Let Ki be the set of all vectors /ceKz, such that 



Jk(p) = for every peT^ . 



Notice that we do not assert that Ki is a linear manifold. 

 If A:eKi then keK^ and (3) above applies. Then 



(vi , ki) + (v, , k.2) = 



and, using this and the PR property of each summand, we conclude 

 that ki and k2 are in Ki . 



