# 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 *F*_{n} : *X* → [0, +∞] be functionals on *X* for each *n* = 1, 2, ... The sequence (or, more generally, net) (*F*_{n}) 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
*x*_{n}∈*X*converging weakly to*x*∈*X*,

- upper bound inequality: for every
*x*∈*X*there exists an approximating sequence of elements*x*_{n}∈*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

## References

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