A very good treatment of the subject can be found in *"Topological Vector Spaces"* by Helmut H. Schaefer.

If you have no previous experience at all with functional analysis, it may be a bit harsh in the beginning though. You can also learn first some more basic functional analysis, for example from the first 3 chapters of the classical book *"Functional Analysis, Sobolev Spaces and Partial Differential Equations"* by Haim Brezis (this is the route I have personally followed for example).

**I expand on my answer in response to the comment**

Note that first that, if $X$ is compact, then all continuous functions on $X$ are bounded. Therefore in this case the "probabilist's weak" topology on the space of finite Radon measures $\mathcal{M}(X)$ is just the weak* topology (in the usual sense of functional analysis) on $C(X)^*$ via the isomorphism $\mathcal{M}(X)\cong C(X)^*$. Therefore the comparison between the two topologies is clear in this case.

This does not work anymore if $X$ is not compact, and here is where the more specific approach of the probabilist comes into play. Let $\mathcal{P}(X)$ be the space of Borel probability on $X$. Then you can define a distance on $\mathcal{P}(X)$ by
$$ d_P(\mu,\nu):=\inf\{\alpha>0\,:\,\mu(A)\le \nu(A_{\alpha})+\alpha,\, \mu(A)\le \nu(A_\alpha)+\alpha\},$$
where
$$ A_\alpha=\begin{cases} \{x\in X\,:\,d(x,A)<\alpha\} & A\ne \emptyset,\\ \emptyset & A=\emptyset. \end{cases} $$
This is called Prokhorov distance, and you can prove that, provided $X$ being separable, the "probabilist's weak" topology is induced by the metric $d_P$.

If fact, in order to make the comparison you are interested in, it suffices to compare this metric with the one inducing the strong topology.

What makes things more complicated from a functional analitic point of view is that the metric $d_P$ is in general not induced by norm. It follows that the functional analysis you most commonly learn in your studies does not apply. Schaefer's book is all about extending those tools to the case in which a norm to start with is not available.

Probabilities and metrics. Convergence of laws on metric spaces, with a view to statistical testing.... or maybe the bookReal analysis and probability$\endgroup$ – Gerald Edgar Jun 17 '19 at 11:58Measure Theory? Section 4.6 discusses the norm topology, and Chapter 8 is all about the weak topology. If it doesn't have what you want, could you be more specific about what it is that you want? $\endgroup$ – Nate Eldredge Jun 17 '19 at 15:16