Appendix to Chapter III
309
(3.3)
x E E and d(a, x) < r' ===} d[b,J(x)] < r.
In this form, the definition extends even to maps into another metric space Y
since it uses only distances. The principle is always the same: the distance
measures both the error in the function and the error in the variable, the error
in the function must be < r so long as the error in the variable is sufficiently
small, Le. < r' in our usual language.
In the case of a Cartesian space, it is clear that for a sequence (un) of
points of JR.P to converge to a limit U it is necessary and sufficient that, for
any i, the ith coordinate of Un should converge to the ith coordinate of u. This
extends trivially (Le. directly from the definitions) to any Cartesian product
Xl x ... X Xn of metric spaces.
In the particular case of JR.P, the general Cauchy criterion remains valid:
for a sequence (un) to converge, it is necessary (in any metric space) and is
sufficient (in JR.P) that, for all r > 0,
(3.4)
d (up, uq) < r for p and q sufficiently large.
To see this, one remarks first that if x and yare two vectors in JR.P, their
: coordinates satisfy IXi - Yil ~ d(x, y) for any i. It is then clear that, for all i,
the coordinates of index i of Un form a Cauchy sequence in JR., so converge to
limits which are clearly the coordinates of the desired vector lim Un.
Though Cauchy's criterion is valid in JR. and so in JR.P, it is not always
, valid in an arbitrary metric space X as already shown by the case of X =
Q endowed with the usual distance. The metric spaces in which Cauchy's
criterion is valid are called complete. JR. is complete, Q is not, which explains
the terminology.
The space CO(K) of continuous functions on a compact interval K c JR.,
endowed with the distance dK(f, g) of uniform convergence, is complete: this
follows from Theorem 13" of Chap. III, nO 10 and the fact that a uniform
limit of continuous functions is continuous (Chap. III, nO 5, Theorem 5).
Likewise, and for more trivial reasons, the space 8(M) of bounded functions
on a set M is complete with respect to the uniform metric on AI. If on the
other hand one endows the space CP(K) considered above with the distance
Ilf - gilK of uniform convergence, one does not obtain a complete space since
a uniform limit of differentiable functions may very well not be differentiable.
To make CP(K) a complete space one has to use the distance (1.7); if a sequence (fn) satisfies Cauchy's criterion relative to the latter, it is clear that
the derived sequences fAil converge uniformly (Cauchy's criterion for uniform
convergence) and Theorem 20 of Chap. III, n° 17, then shows that the sequence (fn) converges in the metric of CP(K).
There is a procedure for "embedding" any metric space X in a complete
metric space X; it generalises the construction of JR. from Q invented by
309
(3.3)
x E E and d(a, x) < r' ===} d[b,J(x)] < r.
In this form, the definition extends even to maps into another metric space Y
since it uses only distances. The principle is always the same: the distance
measures both the error in the function and the error in the variable, the error
in the function must be < r so long as the error in the variable is sufficiently
small, Le. < r' in our usual language.
In the case of a Cartesian space, it is clear that for a sequence (un) of
points of JR.P to converge to a limit U it is necessary and sufficient that, for
any i, the ith coordinate of Un should converge to the ith coordinate of u. This
extends trivially (Le. directly from the definitions) to any Cartesian product
Xl x ... X Xn of metric spaces.
In the particular case of JR.P, the general Cauchy criterion remains valid:
for a sequence (un) to converge, it is necessary (in any metric space) and is
sufficient (in JR.P) that, for all r > 0,
(3.4)
d (up, uq) < r for p and q sufficiently large.
To see this, one remarks first that if x and yare two vectors in JR.P, their
: coordinates satisfy IXi - Yil ~ d(x, y) for any i. It is then clear that, for all i,
the coordinates of index i of Un form a Cauchy sequence in JR., so converge to
limits which are clearly the coordinates of the desired vector lim Un.
Though Cauchy's criterion is valid in JR. and so in JR.P, it is not always
, valid in an arbitrary metric space X as already shown by the case of X =
Q endowed with the usual distance. The metric spaces in which Cauchy's
criterion is valid are called complete. JR. is complete, Q is not, which explains
the terminology.
The space CO(K) of continuous functions on a compact interval K c JR.,
endowed with the distance dK(f, g) of uniform convergence, is complete: this
follows from Theorem 13" of Chap. III, nO 10 and the fact that a uniform
limit of continuous functions is continuous (Chap. III, nO 5, Theorem 5).
Likewise, and for more trivial reasons, the space 8(M) of bounded functions
on a set M is complete with respect to the uniform metric on AI. If on the
other hand one endows the space CP(K) considered above with the distance
Ilf - gilK of uniform convergence, one does not obtain a complete space since
a uniform limit of differentiable functions may very well not be differentiable.
To make CP(K) a complete space one has to use the distance (1.7); if a sequence (fn) satisfies Cauchy's criterion relative to the latter, it is clear that
the derived sequences fAil converge uniformly (Cauchy's criterion for uniform
convergence) and Theorem 20 of Chap. III, n° 17, then shows that the sequence (fn) converges in the metric of CP(K).
There is a procedure for "embedding" any metric space X in a complete
metric space X; it generalises the construction of JR. from Q invented by