FORMAL REALIZAUILITY THEORY — I 253 



12.22 Wc civn now sliow that Ki is a real linear manifold. Consider a 

 real keKi^ilh). Tlien [0, A-]eL(/;n) and by 12.1 



Jdlh) = 0. 



Then l)y 12.1 1 (and ir).r2), Jk{p) = 0, so A'eKi . Since K,j){pn) is spanned 

 by real k (12.21), we have 



K;,,(pn) C K, . 



Comparing- this with (12) gives us 



K,o(po) = Ki . (15) 



Since K/,n(/)n) is a real linear manifold by definition and 12.21, we see 

 that Ki is. 



12.3 Let us now write, by (12) and (15), 



K, = K.> e Ki (16) 



where K^ is an arbitrary fixed manifold disjoint from Ki and with it 

 spanning K^ . All three manifolds are real (12.21, (15), 10.6). 

 Choose a K;i disjoint from K/, such that 



K = K:i e K. e Ki . (17) 



Let the decomposition of V dual to (17) be (10.6) 



V = V3 e Vo e Vi . 



Then V.3 = (Ko KO" = (Kj" = V,.o by 12.01. Hence 



V = V;.n e V2 ® Vi . (18) 



By (14) and the definitions, 



V.0 C Ydp) C V.0 © V2 . (19) 



12.31 Consider a real p„ . Then by P3'(I), (15) and (16) we have 



Kz.o(po) Q K,(pn) C Ko © Kz,o(po). (20) 



This is now an expression dual to (19). We shall prove next that, given 

 any keKL{po)^^2{= K2), there is a unique VkeYLipojC^^i such that 



[v, , k]eL(p,). (21) 



Dually, given any veV L(pn)(^y2 , there is a unique /v,.eK/,(P())nK2 such 

 that 



[r, k„]eL{po). 



