FORMAL REALIZABILITY THEORY — II 561 



(2)/ 



4.46 Multiplying (8) on cither side by Z"{v), 

 -^.GZ''\p)+Y''\p)Z''\p) = 1 



p + Wo 



(9) 

 = -^,Z''\p)G + Z''\p)Y''\p). 



Here, to be strictly correct, we should write two separate equations, 

 interpreting 1 as the identity operator in K for, here, the left ecjuality, 

 and as the identity operator in V for the right equality. Multiplying 

 (9) through b}^ p — iuo and letting /; — ^ iwo , we obtain 



GZ^'\iu:o) = = Z^'\ic^o)G. 



Here, as in (9), we have condensed two dimensionally incompatible 

 equalities. From this it follows that each of G and Z^^\ioio) has its range 

 in the null space of the other. In particular, therefore, the range of G is 

 contained in J. 



4.47 Consider now a v such that Gv = 0. Then, bj^ (7) and (8), 



V ^ Z''\p)Y''\p)v ^ Z''\p)Y''\p)v 

 so, at lojo > 



V = Z^'\ic^o)Y^'\io:o)v = Z^'\iiOo)k 



for some finite vector k = Y^^\io3o)v- Since Z^^^iuo) is finite, v 9^ implies 

 that k 9^ 0. Then, however, v lies in the range of Z^^\icoo). Combining 

 this fact with the result of 4.46, we see that for Gv = it is necessary 

 and sufficient that v lie in the range of Z^"\iwo): the range of Z^^'(zwo) 

 is exactly the null space of G. 



4.48 Now in Halmos , par. 37, it is shown that for any dimensionless 

 operator in an n-space the dimensionality of its range space (its rank) 

 and the dimensionality of its null space (its nullity) add up to 71. A 

 similar result and proof hold for operators between V and K. Let m be 

 the dimensionality of J. Then n — w is the rank of Z^^\icoo), and there- 

 fore the dimensionality of the range of Z^^^iwo), and by 4.47 the dimen- 

 sionality of the null space of G. Hence, finally, 



rank (G) = n — (n — m) = m. 



By 4.46, therefore, J is exactly the range of G. 



4.49 Now N^^\ whose admittance matrix is Y^^\p), might not be ex- 



