FORMAL REALIZAUILITY THEORY — II 559 



(1). Certainly wo may take coo > 0, liocauso /?(zco) = R( — iw). Further- 

 more, by (1), 



Z(/a;„)/.- = il(ic^,)k. (2) 



/(/oj), beiiij^ odd, aiul finite evcrywheic on p = /w, must vanisli at co = 0, 

 and at ice = 0° . Ilenee if wu = or /wu = oo , Z(iajo)/'" = and Z(p) is 

 singular on p = /co. This denies oin- hypothesis that N e (2, 1 + 2). 



4.42 Let J be the set of all \e('tors k eK such that (1) holds: the luUl 

 space of /^(/coo). Then cleail>' J is a linear manifold. Furthermoi-e, J is 

 real, because, if (1) holds then 



R{m^)k = R(im)k = R{im)k = = 



and k also is in J. 



Relations (1) and (2) hold for all A' e J. 



4.43 By its construction, I(iuo) is real and symmetric, but not necessarily 

 definite. There does however exist a real diagonal matrix D and a real 

 non-singular IT such that /(zcoo) = W'DW. Let D+ be the (diagonal) 

 matrix obtained from D by replacing all negative elements of D by zero, 

 and define Z)_ by 



D = D+ - D^. (3) 



Then D+ and Z)_ are real, symmetric, and non-negative semidefinite. 

 Define 



A = o:oW'D+W, 



B = - W'D^W. ^^^ 



We have chosen coo > 0, so A and B are both real, symmetric and non- 

 negative. Certainly therefore 



Z''\p) = Zip) + -A + pB (5) 



V 



is PR. Also Z^'\p) has an inverse, because Z{p) has one by hypothesis 

 and 2.2 applies. 



4.431 Let I' e V be such that for some /:i c K 



V = Aki 

 and for some k-i e K 



V = Bki . 

 Then ?; = 0. 



