When studying different topics in mathematics, such as Probability and Analysis, often new types of convergence show up. For example, Probability talks about convergence in distribution, also calling it sometimes weak convergence, which also appears in Functional Analysis, but there is also weak$*$ convergence, epi-convergence ... Anyways, sometimes things get confusing.
Hence, I was wondering if anyone knows of a “Cheat Sheet” or something similar, which presents and “organizes” all this different types of convergence. When searching on Google, I mostly find things regarding convergence of series, instead of what I’m looking for.
Just to clarify. I’m looking for a reference where the many concepts of convergence (as many as possible) are defined and possibly connected, as the case with weak convergence and convergence in distribution.
Generally speaking, the best cheat sheet for you is the one you write yourself. I can't imagine a better way of reaching an understanding of all the different notions of convergence than to write such a sheet.
However, I will say that the one unifying concept here is that of topology. Each different type of these convergences is really convergence of sequences (or of nets) taking place in some appropriate metric space (or topological space). In probability and analysis, the spaces in question are usually normed vector spaces (or other non-normed topological vector spaces). So if you want to look for a unifying book on the topic, I would look for a book on topological vector spaces.