“Mosco convergence”的意思、由来-开放百科全书

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Mosco convergence

释义

Definition
References
Notes

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.

Mosco convergence is named after Italian mathematician Umberto Mosco, a current Harold J. Gay^[1] 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 → [0, +∞] 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 → [0, +∞] if the following two conditions hold:

lower bound inequality: for each sequence of elements x_n ∈ X converging weakly to x ∈ X,

upper bound inequality: for every x ∈ X there exists an approximating sequence of elements x_n ∈ 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

{{cite journal | last=Mosco | first=Umberto | title=Approximation of the solutions of some variational inequalities | journal=Ann. Scuola Normale Sup. | location=Pisa | volume=21 | year=1967 | pages=373–394 }}
{{cite journal | last=Mosco | first=Umberto | title=Convergence of convex sets and of solutions of variational inequalities | journal=Advances in Mathematics | volume=3 | year=1969 | pages=510–585 | doi=10.1016/0001-8708(69)90009-7 | issue=4 }}
{{cite journal | last=Borwein | first=Jonathan M. |author2=Fitzpatrick, Simon | year=1989 | title=Mosco convergence and the Kadec property | journal=Proc. Amer. Math. Soc.| volume=106 | pages=843–851 | doi=10.2307/2047444 | issue=3 | publisher=American Mathematical Society | jstor=2047444 }}
{{Cite web | last = Mosco | first = Umberto | title = Worcester Polytechnic Institute Faculty Directory | url = http://www.wpi.edu/academics/facultydir/uxm.html | publisher = | accessdate = }}

Notes

1. ^http://www.wpi.edu/Campus/Faculty/Awards/Professorship/gayprofship.html

2 : Calculus of variations|Variational analysis

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