Appendix to Chapter III
Generalisations 48
1 - Cartesian spaces and general metric spaces
In algebra one defines vector spaces over any field, for example Q, lR or C.
The simplest examples, and the most important in analysis, are RP and CP
whose elements, which one calls "points" or "vectors" according to the circumstances, are ordered sequences
of p real or complex numbers, the coordinates of x (Chap. I, nO 4); one speaks
of a real or complex Cartesian space when it is unnecessary to specify the
dimension p. What we call in analysis a "function of several real (or complex)
variables" is just a function, in the general sense of the term, defined on a
subset of a lR P (or CP). We propose, in this appendix, to show very succinctly
how the definitions and results of this chapter extend to these and to other
much more general spaces.
First, inspired by Pythagoras' theorem 49 , we define the Euclidean distance
d(x, Y) 2: 0 between two points of lR P or CP by the formula
(1.1)
on introducing the norm or length
(
2
2) 1/2
IIxll = d(O, x) = Ixd + ... + Ixnl
of a vector, we then have
d(x, y) = Ilx - YII·
Again we have the same triangle inequality
48 This appendix is a reference text for use as needed in the following chapters.
49 For the needs of analysis one might well choose the simpler formula
d'(x,y) = IXI - Yll + ... + Ixp - ypl
given that d'(x,y)/p < d(x,y) < d'(x,y) for all x,y.
Generalisations 48
1 - Cartesian spaces and general metric spaces
In algebra one defines vector spaces over any field, for example Q, lR or C.
The simplest examples, and the most important in analysis, are RP and CP
whose elements, which one calls "points" or "vectors" according to the circumstances, are ordered sequences
of p real or complex numbers, the coordinates of x (Chap. I, nO 4); one speaks
of a real or complex Cartesian space when it is unnecessary to specify the
dimension p. What we call in analysis a "function of several real (or complex)
variables" is just a function, in the general sense of the term, defined on a
subset of a lR P (or CP). We propose, in this appendix, to show very succinctly
how the definitions and results of this chapter extend to these and to other
much more general spaces.
First, inspired by Pythagoras' theorem 49 , we define the Euclidean distance
d(x, Y) 2: 0 between two points of lR P or CP by the formula
(1.1)
on introducing the norm or length
(
2
2) 1/2
IIxll = d(O, x) = Ixd + ... + Ixnl
of a vector, we then have
d(x, y) = Ilx - YII·
Again we have the same triangle inequality
48 This appendix is a reference text for use as needed in the following chapters.
49 For the needs of analysis one might well choose the simpler formula
d'(x,y) = IXI - Yll + ... + Ixp - ypl
given that d'(x,y)/p < d(x,y) < d'(x,y) for all x,y.