Let $X$ and $Y$ two banach space. We say that $L:X\longrightarrow Y$ is a compact operator if $L(U)$ is relatively compact whenever $U$ is bounded.
I recall that a set $E$ is relatively compact if its closure $\bar E$ is compact.
I suppose that if we give such definition for compact operator, it's because there are set that are bounded but s.t. the closure is not compact (otherwise the definition would be that $L:X\longrightarrow Y$ is compact if $L(U)$ is bounded whenever $U$ is bounded), but I can't find such set. So do you have an example of bounded set s.t. the closure is not compact ? If not, why such a definition of compact operator ?