272 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



18.35 Consider P5 (I): if 



[UrJrVMiv), 



where jr is real, r = 1, 2, then 



(Wr , ^)l = {C*Vr , js)l = {Vr , Cjs)l , (5) 



where [tv, Cjr]eL{p). Since Cjr is real 



{vi , Cj-i)i = {v-i , Cji)i 

 by P5(I) for L. This with (5) for r ^ s proves P5(I) for .1/. 



18.36 Fix a jtjjif and for each peW a uip) such that 



Hp),j]eM{p). 



Then u(p) = C*v(p) and 



Hp), Cj]eL{p), 



for some v{p). Then as in (5) above 



(u{p),j)2 = ivip),Cj)r. 



P6(I) and P7(I) for L then imply that P6(I) and P7(I) hold for M, 

 using Tl for r.„ in P6. 



18.37 We now have M satisfying the hypotheses of 12.0. Therefore 

 there is a T m such that M satisfies all the postulates. This is 18.3. 



18.4 Proof of "only if" in 17.2: Suppose that L between V and K is 

 the juxtaposition of Li between Vi and Ki , L2 between V2 and Ko . 

 Let, say, U = Vi and J = Ki . Let C be the identity map from Ki to K: 

 if jej = Ki , then Cj is just j considered as a vector in K. Then C is 

 real. It is easily computed that C* is Ei . 



Consider the restriction M oi L based on this C. Its pairs for 

 P€TmQ:Tl are all the pairs [u, j] such that j = E*jeKL and u = Ev, 

 where 



[v,j]eL{p). (6) 



But then 



[u,j] = [Ev,E*j] 



and this is in Li{p) by (6) and the definition of juxtaposition. Therefore 

 the list M(p) is contained in Li(p). 



Suppose that [u,j]eLi{p). We have [0, 0]eL2(p) so by P2 and the defi- 



