FORMAL REALIZAlJlLirV THEORY — I 243 



(ii) If g(k) is any niimerically values hom()j>;(Mio()us ('f)ii,iuj>;aie-liiioar 

 function of A'eK (i.e., if g{k) is linear in A:) tiien there is a unique ?'„eV 

 such that 



for all keK. 



10.2* The annihilator (Vi)" of a manifold Vi C V is, as in 7.1, the set of 

 all A-6K such that 



reVi implies (v, A:) = 0. 



10.21* It is shown in Ilalmos-' that, to each basis v , • • • , v" in V there 

 exists a unique dual basis A- , • • • , A-" in K such that 



{V, k') = brs , (2) 



where 5^ is the Kronecker symbol: 5,5 = if r ?^ s, 5rr = 1, 1 < ^, s < n. 

 10.22 If 



[v] = [oi , • • • , a„] 



(3) 

 [A-] = [^1, ■■■ .hn] 



are the ?)-tuples representing v and A; relative to a pair of dual bases, 

 then it is easily computed from (1) and (2) that 



{v, k) = i; aA. (4) 



r=l 



Therefore the concrete scalar product of 7.0 is indeed the geometric 

 scalar product here considered, when we restrict our pairs of bases in 

 V and K alwaj^s to be dual in the sense of (2). 



10.23* We shall use the words "coordinate frame" or simply "frame" 

 to denote a pair of dual bases in V and K. Any basis in V (or K) specifies 

 a frame by the imiqueness result quoted above. 



10.24 We shall henceforth deal always with coordinate frames, in fact, 

 ultimately, real coordinate frames, rather than arbitrary pairs of bases. 

 This means in classical language that we are considering as "geometrical 

 ]5roperties" all properties which are preser\'ed under the group of 

 linear transformations which leave the bilinear form (4) invariant. 

 The properties related to physical realizability will turn out to be 

 invariant only under the subgroup of real linear transformations pre- 

 serving (4). 



* Technical paragraph as explained in Section 2.91. 



