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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''. ''Mosco convergence'' is named after Italian mathematician Umberto Mosco, a current Harold J. Gay〔http://www.wpi.edu/Campus/Faculty/Awards/Professorship/gayprofship.html〕 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'' → () 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'' → () 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 : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mosco convergence」の詳細全文を読む スポンサード リンク
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