Mosco convergence

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In mathematics, Mosco convergence is a notion of convergence for functionals, closely related to the notion of Γ-convergence. Mosco convergence is sometimes phrased as "weak Γ-liminf and strong Γ-limsup" convergence, and is named after the Italian mathematician Umberto Mosco.

[edit] Definition

Let X be a topological vector space and let X* denote the dual space to X. Let Fn : X → [0, +∞] be functionals on X for each n = 1, 2, ... The sequence (Fn) 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 xn ∈ 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 xn ∈ 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.

[edit] Reference

  • Mosco, U. (1967). "Approximation of the solutions of some variational inequalities". Ann. Scuola Normale Sup. 21: 373--394. 
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