Let $(\Omega, \mathcal{A}, P)$ be a probability space and $\mathcal{H}$ a real Hilbert space. Consider two $\mathcal{H}$-valued random variables, $X:\Omega\to\mathcal{H}$ and $Y:\Omega\to\mathcal{H}$, which are dependent. Furthermore, let's assume that $\forall \omega\in\Omega$, $X(\omega)\in C\subset \mathcal{H}$ and $Y(\omega)\in C\subset \mathcal{H}$ where $C$ is a ball of radius $\frac{r}{2}$ centered at the origin. Can we say the following,
$$\|X-Y\|^2\leq r^2$$
Of course, if we had two vectors $x$ and $y$ in $C$ then we can bound the norm $\|x-y\|$ by the diameter of $C$. However, I am naive to probability theory and my peer is telling me that I cannot write $\|X-Y\|^2\leq r^2$ because $X$ and $Y$ are functions, not vectors in $\mathcal{H}$. This doesn't make sense to me because they are pointwise (i.e. for all $\omega\in\Omega$) in $C$.
This is an issue of unclear notation, and if you had said more precisely what your notation means the problem might not have arisen. You are both right: $\|X-Y\|$ is a function on $\Omega$, and the inequality $\|X(\omega)-Y(\omega)\|^2\le r^2$ holds for all $\omega$. You meant $\|X-Y\|$ in the pointwse sense, and your peer understood you to mean it was some kind of norm on a space of Hilbert space-valued functions of $\Omega$. Without further explanation both interpretations are reasonable.
You might have said something like $P(\|X-Y\|^2\le r^2)=1$, or even "the random variable $\|X-Y\|$ obeys $P(\|X-Y\|^2\le r^2)=1$" or (more tersely), "$\|X-Y\|\le r^2$ holds pointwise in $\omega$".
There is no absolute truth about mathematical notations: they are just tools we use to help convey our ideas. If your notations are misunderstood it is just like your words being misunderstood; if you find out you have been misunderstood, you fix the problem by rewording.