Unbounded operator

Summary

An unbounded operator is a linear map defined on a linear subspace of a normed space, its domain, with values in another normed space. Despite the name, it need not be unbounded: the term means that boundedness is not assumed. Differential operators such as $f \mapsto f'$ and the position and momentum operators of quantum mechanics are the main examples. The domain is part of the data, and two operators with the same formula but different domains are different.