Metric Spaces, Uniform Boundedness Principle

Uniform Boundedness Principle

Let f be a family of continuous functions from a complete metric space S into R¹. Assume for every x in S, some real number m(x) is an upper bound for the absolute values of all the images f_i(x). There is a nonempty open subset of S, whose images in R¹ are all bounded by some constant m.

For each positive integer m, let U_m,i be the set of x in S with |f_i(x)| ≤ m. Let U_m be the intersection over i of U_m,i. Since each f_i is continuous, each U_m,i is closed, hence each U_m is closed.

By assumption, each x has an m(x) bounding all the images f_i(x), which means each x belongs to some U_m, and S is the countable union over U_m. By the first category theorem, some U_m fails to be nowhere dense. It contains an open set, and the image of this open set is bounded by m.

Perhaps S is not a complete metric space, but the functions in f are all uniformly continuous. These functions can be extended to the completion of S. The uniform boundedness principle applies, And since each f_i is the same on S and on the completion of S, we can pull this result back to S. Therefore, some open set in S has an image bounded in R¹.