Dual and completion of metric spaces

Solution 1:

This might not be the construction that you are expecting in this question. But, let me explain you a possible natural definition for "dual of a metric space". Since all the duality results are secretly rooted in category theory, one should categorify the notion of metric spaces to find the "correct" dual of it. First I should confess you that I am learning some of the notions (in category theory) mentioned in this answer and, therefore would like to keep things elementary as possible.

In this paper, William Lawvere found a way to interpret of metric spaces (or rather generalized spaces) as categories enriched over the monoidal poset category $([0,\infty),\ge,+)$. For more details look at this ncatlab page and this nice YouTube video. But in this setting, Cauchy completion is bit involve and, can be thought of as "the enriched Yoneda embedding" :). Now that we are working with categories instead of metric spaces, we have a natural dual (opposite category) and, it provide us a dual for the initial metric space.