FORMAL Ri: VIJZAIULITY THEORY 1 2o9 



is a PR function (12.11). "Regarding v and k in Vo and K2 let 

 Then by (1) of 7.0 



m m 



(v, ^O = 2 S as,(p)bi+rhs+r 



( = 1 s = l 



and this has a non-negative real part if Ke(p) > 0. This is (iv) of 1.1- 



13.2 AVe can now prove the lemmas 8.1 and 8.2. Given a linear cor- 

 respondence [L] which satisfies PI, • • • , P7 by 11.13 we can interpret 

 [L] as the concrete correspondence representing a geometrical cor- 

 respondence L in some chosen real frame, and L satisfies PI, • • • , P7. 

 Then b}^ the results in 13.01-13.12 there exists a real frame in which the 

 representative [L]i of L has the properties claimed in 8.1 and 8.2 for L w . 

 But we saw in 11.22 that [L] and [L]i are related by a real matrix W like 

 the L and Lw of Section 8. Q.E.D. 



13.21 With the proofs of 8.1 and 8.2 we have reduced the sufficiency 

 claimed for PI, • • • , P7 in 8.0 to the sufficiency of positive reality of 

 [Z(p)] claimed in 1.1, by the argument outlined in 8.5. 



XIV. OPERATOR-VALUED FUNCTIONS OF p 



The next three sections are directed principally toward the proof of the 

 matrix theorem of 1.1. They do however, contribute to 12.10 and to 

 the necessity proof. 



14.0 We continue to use the geometric language. The reader who re- 

 gards this as unduly pedantic is free to place a concrete interpretation 

 upon every argument, for all of the arguments are either frankly based 

 on matrix representations or upon the three identities: 



14.01 {Zj, k) = (Z*k, j) for all j, /ceK. 

 1402 Zk = (Zk) for all keK. 



14.03 Z' = (Z)* = (Z*) 



14.04 These identities are obvious for matrices using 7.0 and 7.2. 

 Geometrically, the first and second define Z* and Z, and the third 

 defines Z' in two ways. The equivalence of these two ways is a theorem 

 based on (10) of 10.33. 



14.05 The symbol Z will always denote an impedance (operator, matrix, 

 scalar), and Y will always denote an admittance. An impedance oper- 



