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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
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40:. Mosco convergence is sometimes phrased as “weak Γ-liminf and strong Γ-limsup” convergence since it uses both the
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468:{\displaystyle \mathop {\text{M-lim}} _{n\to \infty }F_{n}=F{\text{ or }}F_{n}{\xrightarrow{\mathrm {M} }}F.}
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55:, while in infinite-dimensional ones, Mosco convergence is strictly stronger property.
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504:"Convergence of convex sets and of solutions of variational inequalities"
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487:"Approximation of the solutions of some variational inequalities"
51:. In finite dimensional spaces, Mosco convergence coincides with
353:{\displaystyle \limsup _{n\to \infty }F_{n}(x_{n})\leq F(x).}
239:{\displaystyle \liminf _{n\to \infty }F_{n}(x_{n})\geq F(x);}
110: = 1, 2, ... The sequence (or, more generally,
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lower bound inequality: for each sequence of elements
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135: → if the following two conditions hold:
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is named after
Italian mathematician Umberto Mosco.
590:"Worcester Polytechnic Institute Faculty Directory"
260:there exists an approximating sequence of elements
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541:Borwein, Jonathan M.; Fitzpatrick, Simon (1989).
548:Proceedings of the American Mathematical Society
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491:Annali della Scuola Normale Superiore di Pisa
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543:"Mosco convergence and the Kadec property"
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252:upper bound inequality: for every
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83:continuous linear functionals
36:. It is a particular case of
523:10.1016/0001-8708(69)90009-7
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42:weak and strong topologies
617:Convergence (mathematics)
273:, converging strongly to
46:topological vector space
509:Advances in Mathematics
502:Mosco, Umberto (1969).
485:Mosco, Umberto (1967).
607:Calculus of variations
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127:to another functional
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18:mathematical analysis
612:Variational analysis
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22:Mosco convergence
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28:that is used in
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53:epi-convergence
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555:(3): 843–851.
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497:(3): 373–394.
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38:Γ-convergence
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277:, such that
77:denote the
26:functionals
601:Categories
479:References
79:dual space
65:Definition
446:∞
443:→
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394:∞
391:→
333:≤
302:∞
299:→
219:≥
188:∞
185:→
106:for each
435:→
581:2047444
579:
89:. Let
577:JSTOR
383:M-lim
44:on a
69:Let
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567:hdl
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155:to
114:) (
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85:on
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16:In
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583:.
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463:.
460:F
453:M
440:n
428:n
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412:=
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345:)
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339:(
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108:n
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49:X
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