For a normed linear space $X$ and a Banach space $Y$, the set of all compact operator from $X$ to $Y$, which is denoted by $K(X,Y)$, is normed closed in $B(X,Y)$.
Is there a counterexample which shows that the completeness of $Y$ is necessary?
In other words, can be a non-compact operator is equal to (norm)limit of compact operators?