Appendix to Chapter III
323
(8.2)
Vk(f) = sup If(x) I
Ixl::O;k
is the norm of uniform convergence on the compact interval [-k, k]; for a
given k, the relation dk(f,g) = 0 shows that f(x) = g(x) for Ixl ::; k, but not
that f = g; to be sure that f = 9 one must have dk(f,g) = 0 for every k.
Given an f EX, a real number r > 0 and an integer k > 0, let us write
Bk(f,r) for the subset of X defined by the inequality
We remark in passing that dk+l(f,g) ::::: dk(f,g) always, and so Bk+1(f,r) C
Bk(f, r) for all k, f and r; a finite intersection of "balls" with centre f and
of not necessarily equal "radii" thus again contains a ball. In consequence, a
subset U of X is open if and only if, for every fEU, there exist a k and an r
such that Bk(f, r) c U. For a sequence offunctions fn E X, convergence to f
in this topology means simply that, on every interval of the form [-k, k], and
so more generally on every compact set K c 1R, the sequence fn(x) converges
to f(x) uniformly on K: this expresses the fact that every ball dk(f,g) ::; r
with centre f contains fn for all n sufficiently large. This mode of "compact
convergence" is weaker that uniform convergence on IR as we have already
remarked on several occasions.
One could in fact define the topology of CO(IR) - and more generally of
all topological spaces X endowed, as above, with a finite or countable family
of pseudo distances dk(X, y) - by means of a single distance. Such a function
d(x, y) on a set X has values in [0, +oo[ ; if one replaces it by the function
(8.3)
d'(x,y) = cp[d(x,y)]
where cp : 1R+ ---+ 1R+ is continuous, increasing, and satisfies cp(O) = 0, cp(t) >
o for t> 0, cp(s+t) ::; rp(s) +cp(t) for all sand t, one again obtains a distance
on X. It is almost obvious that, for all a E X, every d-ball with centre a
contains a d'-ball with centre a and vice versa. One can thus submit the
given dk to this type of modification without changing the topology (Le. the
open sets) of X. The function cp may be bounded - choose cp(t) = t/(l + t) -,
so one may assume that, for example, dk(X, y) ::; 1 for all x, y E X. The
formula
(8.4)
323
(8.2)
Vk(f) = sup If(x) I
Ixl::O;k
is the norm of uniform convergence on the compact interval [-k, k]; for a
given k, the relation dk(f,g) = 0 shows that f(x) = g(x) for Ixl ::; k, but not
that f = g; to be sure that f = 9 one must have dk(f,g) = 0 for every k.
Given an f EX, a real number r > 0 and an integer k > 0, let us write
Bk(f,r) for the subset of X defined by the inequality
We remark in passing that dk+l(f,g) ::::: dk(f,g) always, and so Bk+1(f,r) C
Bk(f, r) for all k, f and r; a finite intersection of "balls" with centre f and
of not necessarily equal "radii" thus again contains a ball. In consequence, a
subset U of X is open if and only if, for every fEU, there exist a k and an r
such that Bk(f, r) c U. For a sequence offunctions fn E X, convergence to f
in this topology means simply that, on every interval of the form [-k, k], and
so more generally on every compact set K c 1R, the sequence fn(x) converges
to f(x) uniformly on K: this expresses the fact that every ball dk(f,g) ::; r
with centre f contains fn for all n sufficiently large. This mode of "compact
convergence" is weaker that uniform convergence on IR as we have already
remarked on several occasions.
One could in fact define the topology of CO(IR) - and more generally of
all topological spaces X endowed, as above, with a finite or countable family
of pseudo distances dk(X, y) - by means of a single distance. Such a function
d(x, y) on a set X has values in [0, +oo[ ; if one replaces it by the function
(8.3)
d'(x,y) = cp[d(x,y)]
where cp : 1R+ ---+ 1R+ is continuous, increasing, and satisfies cp(O) = 0, cp(t) >
o for t> 0, cp(s+t) ::; rp(s) +cp(t) for all sand t, one again obtains a distance
on X. It is almost obvious that, for all a E X, every d-ball with centre a
contains a d'-ball with centre a and vice versa. One can thus submit the
given dk to this type of modification without changing the topology (Le. the
open sets) of X. The function cp may be bounded - choose cp(t) = t/(l + t) -,
so one may assume that, for example, dk(X, y) ::; 1 for all x, y E X. The
formula
(8.4)