Uniqueness of best approximation in a strictly convex space
= Uniqueness of best approximation in a strictly convex space
If two distinct elements $u,v$ of a <linear subspace> both minimize distance to $f$, normalize their residuals by the common positive minimum $d$. <Strictly convex normed space> geometry gives $\|f-(u+v)/2\|<d$, a contradiction. If $d=0$, both elements equal $f$. This proves uniqueness when a best approximant exists, without asserting existence for a nonclosed subspace.