where 



598 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



Therefore 



[C*0,j] = [0,j]eMip). 



this proves (1). 

 We can now write 



Jm = J21 J20 e Jo (2) 



J2 = J21 e J20 , 

 J20 = J2 n J ji/o , 



(3) 



Jo = J3 J4 , 



J MO = J20 Jo . 



Choosing an arbitrary J5 disjoint from J m , we can write, using (2) 

 and (3), 



J = J5 J21 Jmo, (4) 



where 



Jm = J21 © Jjwo • 



Using the arguments of (I, 12.3), we find that the decomposition of U 

 dual to (4) is, because ilf is PR, 



U = Umo © U21 U3 (5) 



where 



Ua, = \Jmo U21 . 



As in (I, 12.3) we can now introduce a frame appropriate to the de- 

 composition indicated in (4) and (5) and obtain a matrix [Z2i(p)] de- 

 scribing the correspondence between J21 and Uoi . Say this is an m X m 

 matrix, m being the dimension of J21 . We can define 



8(M) = 5([Z2i]). 



Let J2 have dimension ?/ii . By (3), if we border [Zoi(p)] by nii — m 

 rows and columns of zeros, to obtain an nii X Wi matrix [Z^ip)], we can 

 interpret [Z<i(p)] as follows: 



Given i e J2 , it can be represented by an mi-tuple [j] in the basis in 

 that subspace. Then the Wi-tuple 



[«] = [Z2(p)][j] (6) 



