I am given the definition that a locally convex space $V$ is complete if and only if it is isomorphic to its completion $\tilde{V}$.
The completion is constructed in the following way: We have a family of semi-norms $\{p_i\}$ which induce the topology of $V$. For each $i$, we construct a Banach space $B_i=V/K_i$ where $K_i= \{ v\in V\mid p_i(v)=0\}$. Then we take a quotient of the product $\prod_{i}B_i$ in such a way as to guarantee completeness. I may be misunderstanding the process so I apologize if this is not correct.
This definition seems rather abstract, and I can see how it makes sense to define it in this way for generality's sake. But I am struggling to conceptualize how to tell whether a given LCS is complete or not. So my question is: If I am given an explicit family of semi-norms, is there a more hands-on way to check completeness?