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 ones, Mosco convergence is strictly stronger property.
Let X be a topological vector space and let X∗ denote the dual space of continuous linear functionals on X.
The sequence (or, more generally, net) (Fn) is said to Mosco converge to another functional F : X → [0, +∞] if the following two conditions hold: 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