quantum mechanics - Application of the uniform boundedness principle to Physics

The uniform boundedness principle is the following result from functional analysis:

Let $X$ be a Banach space and $Y$ a normed linear space. Suppose that $F$ is a collection of continuous linear operators $T : X\to Y$. If for all $x\in X$ we have

$$\sup_{T\in F}\|T(x)\|_Y<\infty,$$

then

$$\sup_{T\in F, \|x\|_X=1}\|T(x)\|_Y=\sup_{T\in F}\|T\|<\infty.$$

I've heard this theorem is very important in functional analysis. Now, Quantum Mechanics uses a lot of functional analysis, and because of that I wanted to know: what are the implications of this theorem in Quantum Mechanics?

What are the applications of this result inside Quantum Mechanics? Here I'm also counting results directly implied on functional analysis by this result which are of importance in QM.

If not in Quantum Mechanics, is this result used somewhere else in Physics?