FORMAL UEALIZAHILITY THKOKY — II ')\)7 



Caiier eqiiivalout [L]i I)}' the use of ideal transformers. Therefore the 

 partit'uhir 8(L) we have defined — whieh depends for its definition upon 

 a somewhat arbitrary ciioice of coorchnale frame — satisfies (i) of 6.01. 



0.2 Lenuna: Let L be a PR geometrical linear correspondence between 

 K and V, and M another between spaces J and U = J* obtained by 

 restricting L as in (I, 18). Then 



8{M) < 8(L). 



Proof: We use the results and notation of (I, 18). In particular, C is 

 a real constant operator from J to K, ('* its adjoint from V to U, and 

 tJH^ pairs of .V(p) are those pairs 



such that 



u = C*v and [r, Cj] e L(p). 



Choose a frame in V and K which reduces L as in G.l. We recall that 

 J u consists of all vectors j e ] such that Cj e Kl (I, 18.31). Let J2 con- 

 sist of all J e J such that 



Let J:j consist of all j e ] such that 



Cj eK,o. 



Then Jo and J3 are disjoint and both are subspaces of J.w . We can there- 

 fore write 



]m = J2 e j:i e J4, 



after a suitable choice of J4 . 

 We now claim that 



J3®J4CIJ,,0. (1) 



For we have if./ e J.u that, luiiquely, 



j = J2 + jz + ji, 

 \\here ./,• e J,- . Therefore 



(\} - (\h + cu + Cj, 



where bj' construction Cj-i e K2 , (ji e K/.0 , and, necessarily, then 

 Cji = 0. If j-2 = 0, therefore, Cj e K^o and 



[0,Cj]eL{p). 



