Case study I: Uniqueness proofs in approximation theory

An area in which complicated uniqueness proofs feature prominently is approximation theory and, in particular, the topic of best approximation. Here the setting is as follows: Let (X,‖⋅‖) be a real normed linear space and E⊆X be a finite dimensional subspace. An element y _b ∈E is said to approximate x∈X best if $\|x-y_{b}\|=\inf\limits_{y\in E}{\|x-y\|}=:\mbox{dist}(x,E)$ .