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,
- upper bound inequality: for every x ∈ X there exists an approximating sequence of elements xn ∈ 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
![{\displaystyle \mathop {\text{M-lim}} _{n\to \infty }F_{n}=F{\text{ or }}F_{n}{\xrightarrow[{n\to \infty }]{\mathrm {M} }}F.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/882e5e0e2bc43a7da714c95142689ffa44590c16)
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".