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Mosco convergence

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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.

Definition

Let X be a topological vector space and let X denote the dual space of continuous linear functionals on X. Let Fn : X →  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 →  if the following two conditions hold:

  • lower bound inequality: for each sequence of elements xn ∈ X converging weakly to x ∈ X,
lim inf n → ∞ F n ( x n ) ≥ F ( 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
lim sup n → ∞ F n ( x n ) ≤ F ( x ) . {\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

M-lim n → ∞ ⁡ F n = F  or  F n → n → ∞ M F . {\displaystyle \mathop {\text{M-lim}} _{n\to \infty }F_{n}=F{\text{ or }}F_{n}{\xrightarrow{\mathrm {M} }}F.}

References

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