= Best approximation in a normed space
{title2=$E_U(f)=\inf_{p\in U}\|f-p\|$}
= Best approximation
{synonym}
Given $f$ in a <normed vector space> and a <vector subspace> $U$, an element $p_*\in U$ is a best approximation if $\|f-p_*\|=\inf_{p\in U}\|f-p\|$. Existence is automatic for a finite-dimensional <vector subspace>: a minimizing sequence can be bounded by the <triangle inequality>, and a subsequence converges by <compactness> in finite dimension. An arbitrary infinite-dimensional <vector subspace> need not contain a minimizer. The norm matters: <best uniform approximation> uses the <supremum norm>, while $L^1$ approximation minimizes integrated absolute error.
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