242 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



10.01 The effect of attaching a geometric identity to vectors, rather than 

 to M-tuples, is to make it possible to {;hoose coordinate bases freely and 

 as convenient, without elaborate constructions or even interpretations. 

 We can then discuss properties of n-tuples (and other objects, e.g. 

 matrices) which are invariant under the kind of operations exemplified 

 by (1) and (2) of Section 9 as properties of a single geometric object, 

 rather than as properties shared by an extensive class of concrete ob- 

 jects which are converted into each other by the group of operations. 

 10.1 This change in point of view need not change formally anything we 

 have said to date; it simply erects a conceptual superstructure, or pro- 

 vides a conceptual foundation, depending on the reader's personal 

 attitude. 



We shall support this statement by going through the important ideas 

 of Sections 4, 6, and 7 and examining their geometrical meanings or 

 counterparts. It is convenient to consider first and at some length the 

 notions of scalar product and complex conjugate. The geometric struc- 

 ture will then be complete enough to permit a rapid survey of the 

 remaining ideas. 



10.11* The geometrical counterpart of the scalar product introduced in 

 7.0 is a numerically valued function a = a{v, k) of two vector variables. 

 Its first argument v ranges over V and its second argument k ranges over 

 K. The function (x{v, k) is linear in v and conjugate linear in k: 



(x{au -\- hv, k) = aa{u, k) -f ha{v, k), 



a(v, ak -jr bC) - daiv, k) + baiv, (). 



We denote this function (j{v, k) by the simple bracket notation {v, k). 



10.12 With this scalar product, the geometry of V and K is that of a 

 space K and the space K* = V of conjugate linear functionals over K. 

 This is analogous to the real geometry of space and conjugate space 

 discussed at length in Halmos^. In fact, in the introduction to Chapter 

 III of Halmos^ the modifications introduced by the conjugate linearity 

 of {v, k) over K are treated in detail. 



10.13* Because of its importance, we quote here a paraphrase of the 

 results covered in Halmos^, par. 12. 



(i) If f{v) is any numerically valued homogeneous linear function of 

 I'cV, then there is a unique vector A-/eK such that 



f(v) = {v, kf) 

 for all veY. 



* Technical paragraph as explained in Section 2.91. 



