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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 ?

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    Regarding the name, probably the best one can say is that it's statement is its name. If you want to know is how to look the theorem up in a book, I suggest looking in a topology book under a chapter with a title like "metric spaces" or "complete metric spaces" or "completeness in metric spaces" or something like that. – 2017-02-02
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    @LeeMosher Thanks: I have a reference for the proof, https://math.berkeley.edu/~sagrawal/su15_math104/lec17_unifcont.pdf (page 2) it's only the name I was curious about - any idea who first proved it ? – 2017-02-02

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