 Elastic net regularization in Hilbert sequence space

Extend $J$ by infinity outside ${\ell }^1$. For a bounded linear operator $K:{\ell }^2\to {\ell }^2$, the objective $\Vert Ku-f{\Vert }_2^2/2+\alpha J(u)$ is coercive, weakly lower semicontinuous and strictly convex for $\alpha >0$, so it has a unique minimizer. The quadratic penalty ensures existence and uniqueness even if $K$ has a null space. Its limiting exact solution minimizes $J$ in the data fiber, which need not be the minimum-norm least-squares solution.