 Nonlinearity of elastic net reconstruction

For fixed $\alpha >0$ and nonzero bounded $K$, the reconstruction vanishes when $\Vert K^{\ast }f{\Vert }_{\infty }\le \alpha$, which includes a neighborhood of zero. If it were linear it would vanish everywhere. Choose a coordinate $e_j$ with $K{e}_j\ne 0$ and data $f=tK{e}_j$ with $t\Vert K{e}_j{\Vert }^2>\alpha$; the optimality inequality at zero fails, so its minimizer is nonzero. Thus the map is nonlinear. The qualification $K\ne 0$ is essential: $K=0$ gives the zero linear map.