Uniform boundedness principle is a well known and powerful result in functional analysis. This theorem was proved in a paper of Banach and Steinhaus (1927) (Banach is a student of Steinhaus). Some book says their original manuscript was lost during the war, so we do not know how they proved it. Nevertheless, the referee of the paper, Saks, suggested proving it by Baire category theorem, which is a result in general topology. This way of proving is shown in most of the texts and in undergraduate course on functional analysis.

It is believed that Banach and Steinhaus did not prove this by Baire category theorem, but by a so called “gliding hump argument”, which is found in many papers at that time. It is probably first appeared in the work of Lebesgue (1905). Note that “glide” means moving with a smooth quiet continuous motion, and “hump” means a rounded raised mass.

This post shows how to prove this result by the gliding hump technique.

First recall

Let be a family of bounded linear operators from a Banach space to a normed space . If the family is pointwisely bounded, then it is uniformly bounded. In other words, if for all , then .

Suppose the assumption is true but the contrary that . We want to construct sequences in the family and such that for all ,

(1) and

(2) , where .

Indeed, first choose and in the family such that and .

Suppose and are chosen. Set .

Choose in the family so that .

Since , choose such that and .

Hence by (2) and (3)

,

and by (3)

.

Let (exists in as the sequence of partial sums is Cauchy and is complete). Then by (1), (4) and (5)

,

contradicting the hypothesis, so the proof is complete.

Why do people call this kind of argument “gliding hump”? Give me some time (or leave to readers) to figure out it.

Nice post.

I have a question: do you require each in your family to be bounded?

Yes. Corrected.