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I want to consider the Moreau envelope $\psi_j$ of a proper, convex, lower semicontinuous function $\psi$ over a Banach space $V$ on dense (finite dimensional) monotonically increasing subspaces $V_m$, i.e. $V_m \subset V_{m+1}$ and \begin{equation*} \overline{\bigcup_{m\in \mathbb{N}} V_m } = V. \end{equation*}

The Moreau envelope is also proper, convex, lsc and additionally coerciv so the minimization problem \begin{equation*} \min_{v_m\in V_m} \psi_j(v_m) \end{equation*} has a solution $\phi_{jm} \in V_m$ for each $m\in \mathbb{N}$. Now my question: Is the sequence of minimizers $\{\phi_{jm}\}_{m\in \mathbb{N}}$ bounded ( and maybe even independent of $j$)?

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    $\begingroup$ Without additional assumptions, existence of minimizers is not guaranteed (think of linear functionals). You could add, for example, coercivity. $\endgroup$
    – Dirk
    Commented Mar 19, 2017 at 14:42
  • $\begingroup$ Yeah, of course, thanks. I just forgot to mention this. However, the question stays the same. $\endgroup$
    – malwin
    Commented Mar 20, 2017 at 16:30

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