324
Appendix to Chapter III
then obviously defines a true distance 58 on X and the topology of X is
identical to the one obtained using this distance. The inequality dk(X,y) :5
2 k d(x,y) shows that the set d(a,x) < r/2k contains the set dk(a,x) < r.
As on the other hand one has dk (a, x) ::; 1, the relations d1 (a, x) < rl, ... ,
dk(a,x) < rk imply
(remainder of a geometric series). Choosing k sufficiently large and the ri
sufficiently small, one then sees that the ball d( a, x) < r is contained in a
finite intersection of balls di(a, x) < ri, whence the identity of the topologies.
This result applies for example to C°(lR), but do not believe that it makes
this space into a normed vector space: the modification (3) that one has to
make to the functions (1) to make the series (4) converge, though preserving the fact that dk(f, g) depends only on f - g, destroys the homogeneity
property Vk().f) = 1).lvk(f); the function
from which the distance (4) is derived by subtraction, is not an actual norm;
it is easy to show that in fact there is no norm on C°(lR) which, alone, can
define the same topology as the family of functions (1).
The above construction of a topology on C°(lR) can be generalised. To
define a topology on a real or complex vector space E one chooses a family
(Vi)iEI of semi-norms, i.e. of functions with positive values possessing all the
properties of a norm except perhaps the fact that, for i given, the relation
Vi(X) = 0 implies x = o. The functions di(x, y) = Vi(X - y) then define a
topology on E for which the translations x f---> x + a and the homotheties
x f---> ).x are continuous. A neighbourhood of 0 is then a set which contains
all the solutions of a finite number of inequalities of the form Vi(X) < ri, with
ri > o. These topological vector spaces are called locally convex since every
neighbourhood of 0 (or, by translation, of any other point) contains a convex
neighbourhood of 0, this because
for 0 ::; t ::; 1. These are the only topological vector spaces which play a
role in analysis, mainly via the theory of distributions. In particular, the
Hahn-Banach Theorem of nO 6 extends to these spaces.
58 At least if one assumes that dk(X, y) = 0 for all k implies x = y, which will
ensure that d{x, y) = 0 ==> x = y. Geometrically, this means that if x =I- y,
there exists a neighbourhood V of x and a neighbourhood W of Y such that
V n W = 0 [choose an index k such that dk{X, y) = r > 0 and define V and W
by the inequalities dk{X,Z) < r/3 and dk{Y,Z) < rI3). One says then that the
topological space considered is separated, a necessary and sufficient condition for
every sequence to have at most one limit.
Appendix to Chapter III
then obviously defines a true distance 58 on X and the topology of X is
identical to the one obtained using this distance. The inequality dk(X,y) :5
2 k d(x,y) shows that the set d(a,x) < r/2k contains the set dk(a,x) < r.
As on the other hand one has dk (a, x) ::; 1, the relations d1 (a, x) < rl, ... ,
dk(a,x) < rk imply
(remainder of a geometric series). Choosing k sufficiently large and the ri
sufficiently small, one then sees that the ball d( a, x) < r is contained in a
finite intersection of balls di(a, x) < ri, whence the identity of the topologies.
This result applies for example to C°(lR), but do not believe that it makes
this space into a normed vector space: the modification (3) that one has to
make to the functions (1) to make the series (4) converge, though preserving the fact that dk(f, g) depends only on f - g, destroys the homogeneity
property Vk().f) = 1).lvk(f); the function
from which the distance (4) is derived by subtraction, is not an actual norm;
it is easy to show that in fact there is no norm on C°(lR) which, alone, can
define the same topology as the family of functions (1).
The above construction of a topology on C°(lR) can be generalised. To
define a topology on a real or complex vector space E one chooses a family
(Vi)iEI of semi-norms, i.e. of functions with positive values possessing all the
properties of a norm except perhaps the fact that, for i given, the relation
Vi(X) = 0 implies x = o. The functions di(x, y) = Vi(X - y) then define a
topology on E for which the translations x f---> x + a and the homotheties
x f---> ).x are continuous. A neighbourhood of 0 is then a set which contains
all the solutions of a finite number of inequalities of the form Vi(X) < ri, with
ri > o. These topological vector spaces are called locally convex since every
neighbourhood of 0 (or, by translation, of any other point) contains a convex
neighbourhood of 0, this because
for 0 ::; t ::; 1. These are the only topological vector spaces which play a
role in analysis, mainly via the theory of distributions. In particular, the
Hahn-Banach Theorem of nO 6 extends to these spaces.
58 At least if one assumes that dk(X, y) = 0 for all k implies x = y, which will
ensure that d{x, y) = 0 ==> x = y. Geometrically, this means that if x =I- y,
there exists a neighbourhood V of x and a neighbourhood W of Y such that
V n W = 0 [choose an index k such that dk{X, y) = r > 0 and define V and W
by the inequalities dk{X,Z) < r/3 and dk{Y,Z) < rI3). One says then that the
topological space considered is separated, a necessary and sufficient condition for
every sequence to have at most one limit.