MAIN LINES OF MATHEMATICS — COOPER 333 



tire set of axioms for the real numbers cannot be guaranteed to hold 

 for any structures other than the real numbers. 



An effect of this tendency to deal with structures which, because 

 they are simpler in the logical sense, are therefore less narrowly de- 

 fined, is that some important mathematical terms have come to have 

 their meanings extended from the original sphere of reference to cover 

 things having some structure in common with the original notion. 

 A good example is the word "space." The use of this word has been 

 extended to any set of elements which shares the algebraic properties 

 of displacements in Euclidean space — namely, the possibility of being 

 added together and of being multiplied by a number; this gives us 

 the notion of a vector space. On the other hand, the word is used for 

 any set of objects with a topological structure. These two ideas com- 

 bine very fruitfully in the theory of topological vector spaces, and, 

 more specifically, the theory of linear function spaces. By these are 

 meant sets of functions: the "points" of the space are functions, /(a?), 

 g{x), of some variable x/ functions can be added or multiplied by 

 numbers to give other functions. Moreover, we can define the distance 

 between functions, in various ways: for instance we might take the 

 "distance" from f{x) to g{x) to be the maximum value of 

 \f{x) —g{x) I if we are dealing with continuous functions. Alterna- 

 tively, we can define it, for the same, or a wider space of functions, by 



"distance from f{x) to ^ (a?) "={/!/(«) -g{x)Ydx]y^ 

 A space of functions with this last definition of distance has prop- 

 erties very similar to those of Euclidean space ; it differs in having an 

 infinite number of dimensions, but properties like the theorems on 

 parallels, on the sum of the angles of a triangle, Pythagoras's theorem, 

 are as in ordinary Euclidean space ; we can for instance say that two 

 functions are "perpendicular" to one another if / f{x) g{x) dx=0. 



We can transfer notions derived from Euclidean spaces to these 

 spaces ; and this has proved helpful in a number of problems both of 

 pure mathematics and of mathematical physics. For example, when 

 we are considering a vibrating system, such as a stretched string or 

 a bell, the possible forms of displacement of the system are "points" 

 in a function space, in which the "distance" above is connected with 

 the energy of the displacement. The mechanical properties of the 

 system enable us to define a sort of "ellipsoid" in this space ; and the 

 principal axes of the "ellipsoid" are connected with the pure, simple 

 harmonic vibrations of the mechanical system. 



The method of transferring ideas from the examples of mathe- 

 matical structures which we encounter in ordinary mathematical ex- 

 perience to more general examples of these structures is both fruitful 

 and dangerous — dangerous because we may be misled by arguments 



