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 $\Vert f-(u+v)/2\Vert <d$, a contradiction. If $d=0$, both elements equal $f$. This proves uniqueness when a best approximant exists, without asserting existence for a nonclosed subspace.