§5. Differentiable functions of several variables
285
shall assume that the sequence (In) converges simply on U, which makes the
connectedness hypothesis on U redundant, and, in practice, is always verified
from the start. With these hypotheses:
Theorem 23. Let (In) be a sequence of functions of class Cion an open
subset U of JR 2 • Suppose that (i) the fn converge simply on U to a limit
junction f, (ii) the derivatives Ddn and D2!n converge to limit functions
gl and g2 uniformly on every compact K cU. Then f is of class Cl and
Df = (gl,g2), i.e. Dd = gl and D2! = g2, or
(i = 1,2).
We mImIC the proof of Theorem 19, introducing the functions
fpq = fp - Iq· Let us work on a compact disc K c U. Since K is convex
we have
(22.1)
on K, by (21.11). But Dlpq = Dip - Dlq, and since, by hypothesis, the
derivatives converge uniformly44 on K, the uniform norms appearing on the
right hand side of (1) are::; r for p and q large, hence so is the left hand side
since Iz' - zl is majorised by the diameter of K. Since limln(z') exists for
all z', for example at the centre a of K, one also has I/pq(a)1 ::; r for p and q
large, whence, by the usual process of dividing c in four, an inequality
for all z E K
once p, q > N. This means that II/p - IqllK ::; r, whence unilorm convergence
of In on K, by Cauchy's criterion. We extend to the case of an arbitrary
compact set by an argument which will be expounded in Chap. V, n° 6
(Corollary 2 of the Borel-Lebesgue Theorem 45 ) and which presupposes no
more than the definition of open and compact sets.
It remains to show that we obtain the derivatives of 1= Hmln by taking
the limits of those of In. Choose an (a. b) E U and consider the functions x 1-+
In(x, b). They are defined on an open subset of JR, are C 1 , and converge to
x 1-+ I (x, b); their derivatives Dl In (x, b) converge uniformly on every compact
set to the function x 1-+ gl (x, b) since the point (x, b) describes a compact
set in C when x runs through a compact set in R Theorem 19 now assures
44 The values of the D fp are linear maps from R2 into R2, but one can clearly still
speak of uniform convergence since the "norm" of a linear map allows one to
measure the "distance" between two such maps: d(A, B) = IIA-BII. Equivalently,
one can reason with the coefficients of Jacobian matrices.
45 namely: for a sequence of functions defined on U to converge uniformly on all
compacta K c U, it is (necessary and) sufficient that, for all a E U, there exists
a disc D c U with centre a on which the sequence converges uniformly: local
character of compact convergence.
285
shall assume that the sequence (In) converges simply on U, which makes the
connectedness hypothesis on U redundant, and, in practice, is always verified
from the start. With these hypotheses:
Theorem 23. Let (In) be a sequence of functions of class Cion an open
subset U of JR 2 • Suppose that (i) the fn converge simply on U to a limit
junction f, (ii) the derivatives Ddn and D2!n converge to limit functions
gl and g2 uniformly on every compact K cU. Then f is of class Cl and
Df = (gl,g2), i.e. Dd = gl and D2! = g2, or
(i = 1,2).
We mImIC the proof of Theorem 19, introducing the functions
fpq = fp - Iq· Let us work on a compact disc K c U. Since K is convex
we have
(22.1)
on K, by (21.11). But Dlpq = Dip - Dlq, and since, by hypothesis, the
derivatives converge uniformly44 on K, the uniform norms appearing on the
right hand side of (1) are::; r for p and q large, hence so is the left hand side
since Iz' - zl is majorised by the diameter of K. Since limln(z') exists for
all z', for example at the centre a of K, one also has I/pq(a)1 ::; r for p and q
large, whence, by the usual process of dividing c in four, an inequality
for all z E K
once p, q > N. This means that II/p - IqllK ::; r, whence unilorm convergence
of In on K, by Cauchy's criterion. We extend to the case of an arbitrary
compact set by an argument which will be expounded in Chap. V, n° 6
(Corollary 2 of the Borel-Lebesgue Theorem 45 ) and which presupposes no
more than the definition of open and compact sets.
It remains to show that we obtain the derivatives of 1= Hmln by taking
the limits of those of In. Choose an (a. b) E U and consider the functions x 1-+
In(x, b). They are defined on an open subset of JR, are C 1 , and converge to
x 1-+ I (x, b); their derivatives Dl In (x, b) converge uniformly on every compact
set to the function x 1-+ gl (x, b) since the point (x, b) describes a compact
set in C when x runs through a compact set in R Theorem 19 now assures
44 The values of the D fp are linear maps from R2 into R2, but one can clearly still
speak of uniform convergence since the "norm" of a linear map allows one to
measure the "distance" between two such maps: d(A, B) = IIA-BII. Equivalently,
one can reason with the coefficients of Jacobian matrices.
45 namely: for a sequence of functions defined on U to converge uniformly on all
compacta K c U, it is (necessary and) sufficient that, for all a E U, there exists
a disc D c U with centre a on which the sequence converges uniformly: local
character of compact convergence.