Mosco convergence

In mathematical analysis, Mosco convergence 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 topological vector space X.

Mosco convergence is named after Italian mathematician Umberto Mosco, a current Harold J. Gay[1] professor of mathematics at Worcester Polytechnic Institute.

Definition

Let X be a topological vector space and let X denote the dual space of continuous linear functionals on X. Let Fn : X  [0, +∞] 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  [0, +∞] if the following two conditions hold:

\liminf_{n \to \infty} F_{n} (x_{n}) \geq F(x);
\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_{n} \xrightarrow[n \to \infty]{\mathrm{M}} F.

References

Notes

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