FORMAL REALIZAIilLITY THEORY — I 247 



K2 is real, then they all are, and that in this case the bases (14) can be 

 chosen to be real. 



Similar considerations apply to decompositions into more summands: 



if 



V = Vi e V2 e • • • v^ 



then 



K = Ki e K2 © • • • © K„ , 

 where 



V* = K, = n v; = (EV; 



XI. GEOMETRICAL CORRESPONDENCES 



11.0 With the geometr}" of V and K now in hand, we consider the 

 geometric aspects of our network theoretic concepts. 



The definition in Section 4 of general 2/z-pole describes a concrete 

 thing and stands unaltered in our geometric view. The definitions in 

 6.11 of the terminology typified by "N admits [v, k] at frequency p" 

 are unchanged except that we should now explicitly indicate that we are 

 discussing concrete n-tuples of complex numbers by placing brackets 

 around the vector symbols, thus: [v], [k]. In other words, a 2n-pole is 

 described by a concrete relation between /i-tuples. 



11.1* All of the postulates PI, • • • , P7 are stated in a language which 

 now has been given an absolute geometric meaning. In this meaning, 

 PI and P2 describe a geometrical linear correspondence between vectors 

 veW and AeK. This is the geometric counterpart of the concrete 

 notion of a linear correspondence between n-tuples. 



11.11 An impedance matrix, as in 6.3, describes a particularly tightly 

 knit linear correspondence, namely a linear function from K to V. 

 The geometrical counterpart is an impedance operator which for each 

 p is by definition a linear homogeneous function which assigns to each 

 vector A'eK a unique v = Z{p)keW. That is: an operator is afunctional 

 relationship between vectors and as such has a geometric identity. 



11.12 It is easy to prove f that, given an impedance operator Z{p), 

 and given any coordinate bases in V and K respectively, there is a 

 matrix [Z(p)], with elements Zrs{p), 1 < r, s < n, such that relative to 

 these bases the coordinates k^ of a vector k and the coordinates tv of 

 V = Z{p)k are related by (7) of 6.3. We call [Z(p)] the matrix of Z{p) 



* Technical paragraph as explained in Section 2.91. 

 t Cf. Halmos'', par. 26. 



