FORMAL REALIZAHILITY THEORY — II 599 



represents in the dual basis in (J2)° a vector u e IJ21 such that 



[u,j] eM{p). 

 Now this u necessarily is of the form 



u = C*v, (7) 



where 



[v,Cj\eUv). 



But j e J 2 , SO Cj e K2 , so V may be taken to be an element of V2 , with 

 components 



[v] = [Z,(p)][Cj] (8) 



in the basis therein. 



We have bases now in V, K, U, and J, each of which has a set of basis 

 vectors spanning, respectively, V2 , K2 , (J2) , and J2 . By definition of 

 J2 , and by (7) and (8), C operates from J2 to K2 , and C* from V2 to 

 (J2)°. Hence in these respective bases C and C* may be represented by 

 nil X 7711 matrices. In these bases then, from (7) and (8), 



[u] = [C*][v] = [C*][Zdp)][C][j]. 



Comparing this with (6), we have 



[Z2(p)] = [C*][Zi(p)][C]. 



Hence by definitions and 5.26, 



8{M) = 8{[Z,]) < 5([Z0]) = 5(L). 



This is the assertion to be proved. 



6.3 We can now turn to (ii) of 6.01. We follow the synthesis procedure 

 of (I, 19), as modified in the remarks of 3.2. 



Consider a network constructed from x reactive elements, r resistors, 

 and some ideal transformers. As in (I, 19.2), the synthesis of this net- 

 work begins by juxtaposing the r -{- x two poles and the ideal trans- 

 formers, all as separate devices. The correspondence [L] established by 

 this juxtaposition is exhibited in (I, 19.2) as one described by a diagonal 

 matrix [Z^{p)] juxtaposed with one described by certain ideal trans- 

 formers. A frame which reduces this correspondence as in 6.1 can be 

 found by a change of basis wholly within those subspaces in which the 

 ideal transformers operate. Hence the degree 5(L) of this correspondence 

 in exactly 8([Zd]) which, by 5.29, is x. 



Now let [M] be the concrete linear correspondence established by the 



