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.

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,
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[{n\to \infty }]{\mathrm {M} }}F.}

References

  • Mosco, Umberto (1967). "Approximation of the solutions of some variational inequalities". Annali della Scuola Normale Superiore di Pisa. 21 (3): 373–394.
  • Mosco, Umberto (1969). "Convergence of convex sets and of solutions of variational inequalities". Advances in Mathematics. 3 (4): 510–585. doi:10.1016/0001-8708(69)90009-7. hdl:10338.dmlcz/101692.
  • Borwein, Jonathan M.; Fitzpatrick, Simon (1989). "Mosco convergence and the Kadec property". Proceedings of the American Mathematical Society. 106 (3): 843–851. doi:10.2307/2047444. hdl:1959.13/940515. JSTOR 2047444.
  • Mosco, Umberto. "Worcester Polytechnic Institute Faculty Directory".