Given data, approximations in a metric space for moving into a normed vector space isometrically.

Here $f_x(y)$ is approximated by $$x_v = [d(x,K_1),d(x,K_2),....d(x,K_N)]$$ by choosing to consider distances from $x$ to only certain points $K_i$ and these points are selected by clustering in the metric space. I'd like some help in putting this approximation in rigorous mathematical terms, to see if I am missing anything crucial.