In a (Hausdorff) topological vector space one can consider bounded sets: a set is bounded if it is absorbed by any open neighborhood of zero (i.e. a dilated multiple contains ). This specializes to the usual definition of a bounded set in a normed vector space: a set is bounded if it is contained in a ball of some finite radius .
A linear operator between topological vector spaces is bounded if it sends each bounded set in to a bounded set in . For normed spaces, this is equivalent to saying that it sends the unit ball to a bounded set. Between finite dimensional normed spaces, every linear operator is bounded. A linear operator between any two normed linear spaces is bounded iff it is continuous.