Definition
Let X be a topological vector space and let X∗ denote the dual space of continuous linear functionals on X. Let Fn : X → be functionals on X for each n = 1, 2, ... The sequence (or, more generally, net) (Fn) is said to Mosco converge to another functional F : X → if the following two conditions hold:
- lower bound inequality: for each sequence of elements xn ∈ X converging weakly to x ∈ X,
- upper bound inequality: for every x ∈ X there exists an approximating sequence of elements xn ∈ X, converging strongly to x, such that
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
Read more about this topic: Mosco Convergence
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