# 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:

• lower bound inequality: for each sequence of elements xn ∈ X converging weakly to x ∈ X,
${\displaystyle \liminf _{n\to \infty }F_{n}(x_{n})\geq F(x);}$
• upper bound inequality: for every x ∈ X there exists an approximating sequence of elements xn ∈ X, converging strongly to x, such that
${\displaystyle \limsup _{n\to \infty }F_{n}(x_{n})\leq F(x).}$

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

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

## References

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