There is a theorem that a bounded linear transformation from a dense subspace of a normed vector space to a Banach space has a bounded linear extension to the whole space. This seems to be sufficiently important to get a name - the Bounded Linear Transformation (BLT) Theorem.
In large part (it remains to prove linearity), this follows from a similar more general result in Metric spaces that a uniformly continuous function from a dense subset of a metric space to a complete metric space has a uniformly continuous extension.
So, does this more general theorem have a name ?