Best approximation in a normed space ($E_U(f)={\inf }_{p\in U}\Vert f-p\Vert$)

Given $f$ in a normed vector space and a vector subspace $U$, an element $p_{\ast }\in U$ is a best approximation if $\Vert f-p_{\ast }\Vert ={\inf }_{p\in U}\Vert f-p\Vert$. 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.

For $1<p<\infty$ and $n\ge 1$, existence follows from finite-dimensional compactness and uniqueness from strictly convex normed space geometry. Translation by $2\pi /n$ and negative translation by $\pi /n$ preserve the target and the error norm. Uniqueness forces the minimizing trigonometric polynomial to be invariant under the first operation and the negative of itself under the second. Invariance removes every nonzero frequency below $n$; negative invariance removes the remaining constant.

For real functions, let $r=f-p_{\ast }$ and put $\mathrm{sign}(0)=0$. If $\int v\mathrm{sign}(r)=0$ for every $v\in U$, then $p_{\ast }$ is a best approximation in a normed space using the $L^1$ norm. The proof is the pointwise inequality $|r-v|\ge |r|-v\mathrm{sign}(r)$, followed by integration. This is a sufficient criterion even if the residual vanishes on a positive-measure set; a general necessary criterion may choose any subgradient in $[-1,1]$ on that zero set.