256 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



ment of Tl , is a constant. We call this constant the nominal rank of 



12.52 Let Tl consist of all peT,^ where [Zi(p)] has its nominal rank. 

 Then r^, has a finite complement. By the reality result of 12.41, if 

 peTi then peT^ ■ 



It is clear that at any peFt the rank of [Zi(p)] does not exceed its 

 nominal rank. 



12.53 By construction, the vectors Wi(p), • • • , Um(p) all lie in Vi:.(p)nV2 . 

 By the reasoning of 12.33, at any real poeTL they span V2 . Hence the 

 nominal rank of [Zi{p)] is m. Therefore, for any perl , [Zi(p)] has rank 

 m and the Hi(p), ■ ■ • , Umip), lying in V2 , still span V2 . Therefore for all 

 perl 



v^(p)nv2 = Vo . 



By (19), then, 



v,(p) = Vz.0 e Vo = V,, , (30) 



a fixed manifold, for all peT^. 



12.54 It is clear by its construction (cf. Halmos'', par. 26) that [Zi{p)] 

 describes the mapping of 12.32 from K2 to V2 = VL(p)nV2 by 



[iv] = [Z^(p)][k]. 



Here the m-tuples [vk] and [A] are the components of Vk and k relative to 

 the bases now available in V2 and K2 . 



12.55 We repeat 



Ki C K,o(p) C K, = Ki e Ko . (12) 



Fix a peTL and a A:€Kz.o(p)nK2 . Then [0, k]eL{p). Since 0€V2 , it follows 

 from 12.54 that [Zi(p)] annihilates A:. Suppose m 9^ 0. Since the rank of 

 [Zi(p)] is m, it follows that A- = 0. Hence for peFi, 



Kz.o(p)nK2 = 0. 



By (12), then, (31) 



K,o(p) = Ki = K^ , 



a fixed manifold. This, with the result of 12.53, proves that L satisfies 

 P3(A), when m 5^ 0. 



If m = then V. = 0, K2 = and (31) follows from (12) and (16). 



12.56 [Zi(p)] is of dimension m and rank m for any ptV^ . Therefore 



