4
1. Selected Problems in One Complex Variable
year graduate course on the subject. Although the ideal theory problem
posed here provides motivation for these results, they have much broader
applicability.
1.3 Partitions of Unity
Proving Proposition 1.2.2 is a typical example of what we will call a local
to global problem. That is, we know that the equation (1.2.1) has local
solutions in the sense that for each w £ U there is a neighborhood V of w
and a solution to (1.2.1) consisting of functions fa holomorphic o n K In
fact, for some j , gj does not vanish at w. Then Vj = {z G U : gj(z) ^ 0} is a
neighborhood of u on which equation (1.2.1) has a solution, given by setting
fj
9J1 a n
d fa = 0 for i ^ j . Thus, we will have proved Proposition 1.2.2 if
we can show that: if equation (1.2.1) has a solution locally in a neighborhood
of each point of [/, then it has a global solution.
We will encounter many of these local to global problems in the course
of our study. Proposition 1.2.2 is a special case of a more general result
concerning a system of linear equations
(1.3.1) GF = H,
where U is an open set in
Cn,
G is a given p x q matrix with entries from
H(U), H is a given p vector of functions from H(U), and a solution F is
sought which is a q vector of functions from H(U). Is it true that, if this
system of equations has a solution locally in a neighborhood of each point
of £/, then it has a global solution on U? The answer is "yes", provided U is
what is called a domain of holomorphy. To prove this result requires much
of the machinery that we shall develop in this text. Every open set in C is a
domain of holomorphy and so the answer is always "yes" for functions of a
single variable. We won't prove that in this chapter, although we will prove
it in the special case of equation (1.2.1).
While the local to global problem posed by (1.3.1) is quite formidable
for holomorphic functions on an open set U in
Cn,
the same problem for the
classes of continuous or infinitely differentiable functions is actually quite
easy. This is due to the fact that the classes of continuous and infinitely dif-
ferentiable functions have a strong separation property - Urysohn's lemma.
Urysohn's lemma for continuous functions on a locally compact Hausdorff
space should be familiar to the reader. A similar result holds for C°° func-
tions on Euclidean space.
1.3.1 Lemma. If K C U C
IR71,
with K compact and U open, then there
exists f e
C°°(IRn)
such that 0 f(x) 1 for all x, f(x) = 1 for x e K,
and f(x) 0 for x £ U.
Previous Page Next Page