Mosco convergence

In mathematical analysis, Mosco convergence is a notion of convergence for functionals that is used in nonlinear analysis and set-valued analysis. Named after the Italian mathematician Umberto Mosco, it is a particular case of Γ-convergence. Mosco convergence is sometimes phrased as “weak Γ-liminf and strong Γ-limsup” convergence since it uses both the weak and strong topologies on a topological vector space X. In finite dimensional spaces, Mosco convergence coincides with epi-convergence, while in infinite-dimensional spaces, Mosco convergence is a strictly stronger property.

Definition

Let X be a topological vector space and let X^∗ denote the dual space of continuous linear functionals on X. Let F_n : X → [0, +∞] be functionals on X for each n = 1, 2, ... The sequence (or, more generally, net) (F_n) is said to Mosco converge to another functional F : X → [0, +∞] if the following two conditions hold:

• lower bound inequality: for each sequence of elements x_n ∈ X converging weakly to x ∈ X,

$\underset{n\to \mathrm{\infty }}{\mathrm{lim\; inf}}{F}_n(x_n)\ge F(x);$

• upper bound inequality: for every x ∈ X there exists an approximating sequence of elements x_n ∈ X, converging strongly to x, such that

$\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}{F}_n(x_n)\le F(x).$

Since lower and upper bound inequalities of this type are used in the definition of Γ-convergence, Mosco convergence is sometimes phrased as “weak Γ-liminf and strong Γ-limsup” convergence. Mosco convergence is sometimes abbreviated to M-convergence and denoted by

${\text{M-lim}}_{n\to \mathrm{\infty }}{F}_n=F\text{ or }{F}_n\underset{n\to \mathrm{\infty }}{\overset{\mathrm{M}}{\to }}F.$