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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



(11) 



two frames, and (b) relates these n-tuples as follows: 



Ml = [wr'[v], 



[k]i = [W]*[kl 



It is easy to show that if [W] has real elements, so that [IF]* = [W]', 

 then the two frames involved above are either both real, or else neither is 

 real. Also, conversely, if both frames are real, then necessarily the [W] of 

 (11) has real elements and [IF]* = [IF]'. 



10.6* Some further important geometrical notions must be mentioned 

 before we proceed. 



If Vi and V2 are disjoint linear manifolds in V — i.e. linear manifolds 

 having in common only the single vector — we write 



Vi e V2 



for the linear manifold consisting of all vectors v = ^1 + ^2, where 

 VieYi , i = 1, 2. The circle around the plus sign is used to denote the 

 disjointness of Vi and V2 . 



It is shown in Halmos^, par. 19, that if 



V = Vi © V2 



then 



(12) 



(13) 



K = K: © K2 , 



where Ki = (¥2)°, K2 = (Vi)" and the dimension of Kj is equal to that of 

 Vi , i = 1, 2. We call (13) the decomposition dual to (12). We some- 

 times write Ki = Vf to denote the Ki dual to Vi in the decomposition 

 (13). It is shown in Halmos^, loc. cit., that there exists a basis V\ , • ■ • , 



Vn in V and its dual ki , ■ 

 ofVi, 



ki, 



kr+l , 



Furthermore, if fi , • • • 



kn in K such that, if r is the dimension 





is a basis for Vi 

 is a basis for V2 

 is a basis for Ki 

 is a basis for K2 



(14) 



Vn is any basis in V satisfying the first half 

 of (14), its dual basis satisfies the second half, and dually. 



We shall show in the Appendix that if any one of Vi , V2 , Ki , or 



* Technical paragraph as explained in Section 2.91. 



