= Strict convexity separates uniqueness from existence in best approximation
A strictly convex <norm> permits at most one <best approximation> from a <linear subspace>, but does not force the distance <infimum> to be attained. In the <Hilbert space> $\ell^2$, the finite-support <vector subspace> $c_{00}$ is dense and omits $u=(2^{-j})_{j\geq1}$. Its distance from $u$ is zero and no member attains that distance. Finite-dimensionality restores <existence> by <compactness> of bounded sets in the <vector subspace>. If $Y$ is a bounded <projection> onto a <vector subspace>, the estimate $\|u-Yu\|\leq(1+\|Y\|)\inf_q\|u-q\|$ follows directly by writing $u-Yu=(I-Y)(u-q)$ for each candidate $q$, even when the <infimum> is not attained.
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