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.