244 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



10.3* Conjugation is an operation which to each vtW associates a 

 vector V uniquely determined by v with the properties 



V = V, 



(5) 



{au + hv) = au + hv, 

 where a and h are any complex numbers and f7, h their conjugates. 



10.31* Given any such conjugation operation in V, and given any 

 A;tK, define a function gkiv) by 



g,{v) = (^) (6) 



for yeV. Then gk(v) is linear in v, by (5) above and (1) of 10.11. There- 

 fore, by 10.13, there is a unique vector keK such that 



Ukiv) = {v, k). (7) 



10.32* Directly from (1) of 10.11 and (6) above, if./ = ah + h(, then 



Qjiv) = agk{v) + hgi{v). 

 From (7), therefore 



(y,j) = a{v,k) + h{v,0 



for all yeV. Comparing this with (1) of 10.11, we see that 



] = ok + hL (8) 



The second item of (5) above then holds for vectors A:eK. 



That % = k follows easily: We have from (6) and (7), written for the 

 vector k, that 



(^) = {v, k). (9) 



We also have, by writing (G) and (7) for vectors v and k that 



(y, k) = (y, k). 



Taking complex conjugates of these two numbers, and using v = v 

 from (5), we have 



(y, k) = (^). (10) 



Then (9) and (10), which hold for all ytV, identify k and k. by 10.13. 



10.34* We have now showed in (5), (8) and (10) that this complex 

 conjugate satisfies the formal properties of the conjugate for n-tuples 

 introduced in 7.2. 



Technical paragraph as exphiined in Section 2.91. 



