= Nonlinearity of elastic net reconstruction
For fixed $\alpha>0$ and nonzero bounded $K$, the reconstruction vanishes when $\|K^*f\|_\infty\leq\alpha$, which includes a neighborhood of zero. If it were linear it would vanish everywhere. Choose a coordinate $e_j$ with $Ke_j\ne0$ and data $f=tKe_j$ with $t\|Ke_j\|^2>\alpha$; the optimality inequality at zero fails, so its minimizer is nonzero. Thus the map is nonlinear. The qualification $K\ne0$ is essential: $K=0$ gives the zero linear map.
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