= Elastic net regularization in Hilbert sequence space
{title2=$J(u)=\|u\|_1+\beta\|u\|_2^2,\quad\beta>0$}
Extend $J$ by infinity outside $\ell^1$. For a <bounded linear operator> $K:\ell^2\to\ell^2$, the objective $\|Ku-f\|_2^2/2+\alpha J(u)$ 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>.
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