270 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



By comparison with (1), we have 



C*v = C*v. 

 Hence C* also takes real vectors into real vectors and is real. 



18.2 Now let L be a PR correspondence between V and K. We define 

 one, say AI, between U and J, as follows: For each peFi, , let M{p) 

 consist of all pairs 



[u, j] 

 such that u = C*v and 



[v, Cj]eL{p). 



This definition can be illustrated l^y enlarging the chart of 18.11 : 



c*l ]c 



The u's corresponding to jej can be constructed by going around 

 through C, L and C*. This then defines a direct mapping from J to U. 



18.21 We call the M defined by 18.2 a restriction of L, since its pairs 

 are images under C* and CT^ (which is not defined over all of K) of a 

 restricted set of pairs drawn from L. 



18.22 Clearly there is a dual operation defined by an operator D from 

 U to V. We might distinguish the operation of 18.2 by calling it a 

 current restriction, its dual by calling it a voltage restriction, 



18.23 The restriction Af of L is defined by lists M(p) which exist for 

 any peFi, . The frequency domain of M has not yet been specified, 

 however. 



18.3 Theorem: If L is PR, then there is a frequency domain T m ior M 

 such that M is PR. 



Proof: PI and P2 for M are evident at once, for any peT^ . The 

 remainder of the proof is divided among 18.31, • • • , 18.37 below. 



18.31 For P3, let Jm be all jej such that CjcKl . Then, given jcJm , 

 for each peTi, there is a y such that 



[v, Cj]eL(p), 

 whence 



[C\j]eM(p). 



