254 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



The proof is a standard one in algebra and depends only upon P2, 



(19), and (20). 



Proof: Given A;eKi,(po)nK2 , there is some yeVz,(po) such that 



[v, k]eL(po). (22) 



By (19), then, 



V = Vo -\- V2 



where VqcVlo , i^2eV2 . Then 



[vo , 0]eL(po) 



SO, applying P2 to this and (22), 



[v - vo,k - 0] = [V2 , k]eL(po). (23) 



Hence i'2eVi(po)nV2 and Vk = v^ satisfies (21), Suppose now 

 v^€Nl{v^)^^2 and 



[vz , k]eL{po). 



Then using this with (23) and P2 



[V2 — Vz , 0]eL(po). 



Hence {v2 — Vi)eWLo . Now VL(po)nV2 is a linear manifold and contains 

 V2 , Vs . Hence 



(^2 - y3)6Vz.onv^(po)nv2 = 0. 



Therefore v^ = Vz . 



The dual argument completes the proof. 



12.32 The argument actually exhibited in 12.31 uses only P2 and (19), 

 hence the Vk of (21) is unique whether or not po is real. Indeed, this is 

 true even when AcKl . 



12.33 The result of 12.31 establishes a bi-unique linear mapping between 

 K2 and VL(7Jo)nV2 . Hence these two manifolds are of the same dimen- 

 sion. Since K2 and Vo = K* are of the same dimension by construction, 

 it follows that 



Vz,(po)nV2 = V2 



and, by (19), that 



Vj,(7>o) = V^o e V2 . 



12.4 Let us now introduce a real frame in V and K which provides real 

 bases in Ki , K2 , K3 and in Vi , V2 , Vlo of (17) and (18). Let ki , • • • ,km 



