I know that a linear operator is bounded iff continuous, with these definitions of "bounded" and "continuous":
a linear operator T (on H Hilbert space) is bounded if ∃M>0: ||Tf||≤M||f|| for all f in H;
a linear operator T is compact if T(B) has compact closure, with B unitary closed ball (or a linear operator T is compact if every bounded sequence {xn} has a subsequence {xnk} such that {T(xnk)} converges).