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  "body": "\n## Definition\n\nLet $X$ be a real [[normed space|03J5RAKB30]].\nWrite $x_n \\rightharpoonup x$ for convergence in the [[weak topology|P6KHVSTPZV]] $\\sigma(X, X^*)$ and $x_n \\to x$ for convergence in norm, where $X^*$ is the [[continuous dual space|4A82S5FVRT]] of $X$.\nLet $F_n \\colon X \\to [-\\infty, +\\infty]$ for $n \\in \\N$ and $F \\colon X \\to [-\\infty, +\\infty]$ be functions with values in the [[extended real line|0GSZ7B0CGR]].\nThe sequence $(F_n)$ *Mosco converges* to $F$ if the following two conditions hold for every $x \\in X$.\n\n1. (Weak lower bound.) For every sequence $(x_n)$ in $X$ with $x_n \\rightharpoonup x$,\n$$F(x) \\le \\liminf_{n \\to \\infty} F_n(x_n).$$\n2. (Strong recovery sequence.) There is a sequence $(x_n)$ in $X$ with $x_n \\to x$ and\n$$F(x) \\ge \\limsup_{n \\to \\infty} F_n(x_n).$$\n\nThe indicator function of a set $K \\subseteq X$ in convex analysis is $0$ on $K$ and $+\\infty$ outside $K$.\nSubsets $K_n \\subseteq X$ Mosco converge to $K \\subseteq X$ if their indicator functions do.\n\n## Remarks\n\nFor sets the two conditions read as follows.\nIf $n_k$ is increasing, $x_k \\in K_{n_k}$ and $x_k \\rightharpoonup x$, then $x \\in K$.\nEvery $x \\in K$ is the norm limit of a sequence with $x_n \\in K_n$.\nThe notation $F = \\operatorname{M\\text{-}lim}_n F_n$ is common.\nIf $X$ is finite-dimensional, weak and norm convergence agree, and Mosco convergence is [[Γ-convergence|NW9MM370A5]].\n\nExample: let $F$ be convex and lower semicontinuous, and let $F_n = F$ for all $n$.\nThen $(F_n)$ Mosco converges to $F$.\nThe recovery sequence $x_n = x$ gives clause 2.\nFor clause 1, a convex lower semicontinuous function is also weakly lower semicontinuous, because its closed sublevel sets are convex and hence weakly closed.\n\nNon-example: in the sequence space $\\ell^2$ with unit vectors $e_n$, let $K_n = \\set{e_n}$ and $K = \\set{0}$.\nClause 1 holds, since $e_n \\rightharpoonup 0$.\nClause 2 fails at $0$: every choice $x_n \\in K_n$ is $x_n = e_n$, and $\\norm{e_n - 0} = 1$ for all $n$.\n",
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