FORMAL REALIZABILITY THEORY — ^I 2G9 



17.31 The labor of coii.sidoring PR con-espondences instead of matrices 

 lias yielded the disappointingly simple resnlt of 17.3. We have already 

 been warned of this, however, by our knowledge of the properties of 

 physical networks (2.9). 



XVIII. THE OPERATION OF RESTRICTION 



18.0 In addition to juxtaposition, which is an operation on correspond- 

 ences clearly motivated by physical considerations, there is an operation, 

 Ikm'c called restriction, which has important use in the next section. 

 There the physical meaning of the operation will become clear. 



18.1 Let V and K = V* be a pair of dual spaces. Let U and J = U* be 

 another pair. Suppose that C is a given fixed linear operation from J to 

 K: given any je], there is a unique k(j)eK, written 



Hj) = Cj, 



such that if Av = QV , r = 1,2, then 



Giki + aoA-o = C(aiji + Uzji) 



for any complex scalars ai , ao . 



18.11 Let {v, k)i denote the scalar product between V and K, and 

 (;/, j)2 that between U and J. Given C, and any veY, let us find that 

 unique vector u{v)e\J for which 



(uiv),jh = (v,Cj), (1) 



for every je]. That such a vector u{v) exists and is unique follows from 

 10.13 when we notice that the right-hand side of (1) defines a function 

 conjugate linear in J. Now for fixed j, the right-hand side of (1) is linear 

 in i\ hence so also is the left side. That is, there is a linear operation C* 

 from V to U such that 



u{v) = C*v. 

 The following chart illustrates the situation : 



V K 



c*[ ]c 

 u J 



18.12 We suppose now that C takes real j into real k, i.e., that C is 

 real. Then by (1) 



{C*v,jh = (C*v,j), = {v,C% = {v,Cj\. 



