FORMAL RF.ALIZABILITV THEORY— I 237 



that 



= (//, A-i - A-,) = {u, /m) - (a, /C2). 



(^IvD. A dual result follows from P3(I). 



7.71* The result of 7.7 above means that the scalar product (v, k) is 

 fixed by r alone when we know that [v, k]€L{p). This means that, given 

 reV,. , there is a unique function Fi.(p) defined for peVi, by 



FM = (v, k) 

 where [v, k]eL{p). Duall.y, 



'hip) = (v, k) 

 is defined for each fixed keKi, . 

 7.72* (PO.) The complement of F/, is finite and 



(I) For each veV^ , Fy(p) is rational 

 (A) For each keK^ , Jk{p) is rational. 

 7.73* (P7.) (A) Re(p) > implies Re(F,(p)) > 

 (I) Re(p) > implies Re(.A.(p)) > 0. 



VIII. THE FUNDAMENTAL REALIZABILITY THEOREM 



8.0* We can now state our fundamental realizability theorem: If a 

 linear correspondence L satisfies PI, • • • , P7, the associated 2/i-pole 

 Nz, is physically realizable. Conversely, given a physically realizable 

 2/i-pole N, the associated linear correspondence satisfies PI, • • • , P7. 



8.01 Actually, the postulates PI, • • • , P7 are not unique nor even en- 

 tirely independent. Many changes may be rung on them. We indicated 

 one above. At the expense of apparent asymmetry, the (A) or (I) por- 

 tions, in various combinations, can be deleted or weakened. We shall 

 not pursue this subject further at this point, but must come back to it 

 in Section 12. 



8.02 We close this Section by outlining the proof of 8.0. The details are 

 then contained in the remainder of the paper. 



8.03 The proof that PI through P7 are necessary for physical realiza- 

 bility will be a direct one: it will be show^n that, considered individually, 

 each network branch and each ideal transformer satisfies the postulates. 



* Technical paragraph as exj)hiine(i in Section 2.91. 



