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''. In finite dimensional spaces, Mosco convergence coincides with epi-convergence, while in infinite-dimensional ones, Mosco convergence is strictly stronger property.

''Mosco convergence'' is named after Italian mathematician Umberto Mosco, a current Harold J. Gayhttp://www.wpi.edu/Campus/Faculty/Awards/Professorship/gayprofship.html 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'' →  , +∞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'' →  , +∞if the following two conditions hold:

* lower bound inequality: for each sequence of elements ''x''_''n'' ∈ ''X'' converging weakly to ''x'' ∈ ''X'',

::$\liminf_ F_ (x_) \geq F(x);$

* upper bound inequality: for every ''x'' ∈ ''X'' there exists an approximating sequence of elements ''x''_''n'' ∈ ''X'', converging strongly to ''x'', such that

::$\limsup_ F_ (x_) \leq F(x).$

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_ F_ = F \text F_ \xrightarrow \to \inftyF.

References

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Notes

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Calculus of variations
Variational analysis
Convergence (mathematics)