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Mosco convergence

In mathematical analysis, Mosco convergence is a notion of convergence for functionals that is used in nonlinear analysis and set-valued analysis. Named after the Italian mathematician Umberto Mosco, 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 spaces, Mosco convergence is a strictly stronger property.

Definition

Let X be a topological vector space and let X<sup>∗</sup> denote the dual space of continuous linear functionals on X. Let F<sub>n</sub>&nbsp;:&nbsp;X&nbsp;→&nbsp;[0,&nbsp;+∞] be functionals on X for each n&nbsp;=&nbsp;1, 2, ... The sequence (or, more generally, net) (F<sub>n</sub>) is said to Mosco converge to another functional F&nbsp;:&nbsp;X&nbsp;→&nbsp;[0,&nbsp;+∞] if the following two conditions hold:

  • lower bound inequality: for each sequence of elements x<sub>n</sub>&nbsp;∈&nbsp;X converging weakly to x&nbsp;∈&nbsp;X,
:
  • upper bound inequality: for every x&nbsp;∈&nbsp;X there exists an approximating sequence of elements x<sub>n</sub>&nbsp;∈&nbsp;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