I'm a first year grad student. I feel that practice problems/examples have been one of the most effective ways to enhance learning of new material. However, there are often no "problem set" for more advanced topics (e.g. papers in a specialized research area). I think if it might be beneficial to make up some problems of my own to enhance my understanding but that sometimes can be time consuming. I'm curious if mathematicians here have any advice for me on this regard.
For those who think this is too broad or off-topic, I'd like to give an example; Suppose someone proposes the following inequality (Schwarz inequality): $\left| \sum_{i=1}^n u_i \bar{v}_i \right|^2 \leq \sum_{j=1}^n |u_j|^2 \sum_{k=1}^n |v_k|^2$ in their new research. I don't see nothing interesting at first glance. I'm curious of any method/tricks that will help me best "absorb the result" without someone gives me a bunch of examples as illustration (because often times those are either scarce or not available for more obscure result)?