Mosco convergence

In mathematical analysis, Mosco convergence, named for the Italian mathematician Umberto Mosco, is a notion of convergence for functionals that is used in nonlinear analysis and set-valued analysis. 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 a topological vector space X.

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,

$\liminf_{n \to \infty} F_{n} (x_{n}) \geq F(x);$

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

$\limsup_{n \to \infty} F_{n} (x_{n}) \leq F(x).$

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

$\mathop{\text{M-lim}}_{n \to \infty} F_{n} = F \text{ or } F \xrightarrow[n \to \infty]{\mathrm{M}} F.$

References

Mosco, Umberto (1967). "Approximation of the solutions of some variational inequalities". Ann. Scuola Normale Sup. (Pisa) 21: 373–394.
Mosco, Umberto (1969). "Convergence of convex sets and of solutions of variational inequalities". Advances in Mathematics 3 (4): 510–585. doi:10.1016/0001-8708(69)90009-7.
Borwein, Jonathan M.; Fitzpatrick, Simon (1989). "Mosco convergence and the Kadec property". Proc. Amer. Math. Soc. (American Mathematical Society) 106 (3): 843–851. doi:10.2307/2047444. JSTOR 2047444.