Let
Γ
(
X
)
\Gamma (X)
denote the proper, lower semicontinuous, convex functions on a Banach space
X
X
, equipped with the completely metrizable topology
τ
\tau
of uniform convergence of distance functions on bounded sets. A function
f
f
in
Γ
(
X
)
\Gamma (X)
is called well-posed provided it has a unique minimizer, and each minimizing sequence converges to this minimizer. We show that well-posedness of
f
∈
Γ
(
X
)
f \in \Gamma (X)
is the minimal condition that guarantees strong convergence of approximate minima of
τ
\tau
-approximating functions to the minimum of
f
f
. Moreover, we show that most functions in
⟨
Γ
(
X
)
,
τ
a
w
⟩
\langle \Gamma (X),{\tau _{aw}}\rangle
are well-posed, and that this fails if
Γ
(
X
)
\Gamma (X)
is topologized by the weaker topology of Mosco convergence, whenever
X
X
is infinite dimensional. Applications to metric projections are also given, including a fundamental characterization of approximative compactness.