252 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



by P2, where [vr(p), kr]€L(p), r = 1, 2. Then (9) holds for each of 

 Vi(p), Vi{p) and therefore also for the v{p) of (10). 



We have showed now that for any peFz, and any AeKi , the vip) of 

 (4) has a vanishing scalar product with every real jeKz, . Since Kl 

 is spanned by real j, 



v{p)e{}Lj)' = W,o. (11) 



12.14 By (11), 



[v(p), 0]eL{p). 

 Comparing this with (4), and applying P2, 



[v(p) - v{p), k-0] = [0, k]eL(p). 

 Since k is now any vector in Ki, we have 



Kl C K^o(p) C K^ (12) 



for every^peTi, . 



12.15 We can now also show that Vi,(p) C (Ki)°. We return to 12.13 

 and read (9) thereof as originally derived for real j and k. Applying 

 P5(I), we have from (9) that 



(m(p), k) = for peVr. . (13) 



By the argument immediately following (9), (13) also holds for any 

 fceKi , provided j is real. As in 12.11 any jeK^ can be written 

 j = jx + iji , where ji and ^2 are real, and the corresponding 



u{p) = ui(p) + iihip) 



where [Ur{p), jV]eL(p). Therefore, finally, (13) holds for any u(p) 

 satisfying (4) — i.e., any w(?>)eV/.(p) — and any keKi . Therefore 



V.(p) ^ (KO" (14) 



for any peT^ . 



12.2 We now fix our attention on a specific real Po^Tl 



12.21 By P4, if 



[v, k]eLipo) 



we have also 



[v, k]eL{po) = L{po). 



In particular, Kim(po) is real. 



