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  "body": "\n## Statement\n\nLet $X$ be a [[normed vector space|03J5RAKB30]] over $\\R$ or $\\C$.\nLet $(x_n)_{n \\ge 1}$ be a sequence in $X$ that converges to $x \\in X$ in the [[weak topology|P6KHVSTPZV]].\nFor each $n \\ge 1$ there are an integer $N(n) \\ge n$ and real numbers $\\lambda_{n,k} \\ge 0$ for $n \\le k \\le N(n)$, with $\\sum_{k=n}^{N(n)} \\lambda_{n,k} = 1$, with the following property.\nThe convex combinations\n$$y_n = \\sum_{k=n}^{N(n)} \\lambda_{n,k}\\, x_k$$\nconverge to $x$ in norm.\n\n## Remarks\n\nFix $n$ and let $C_n$ be the norm closure of the convex hull of $\\set{x_k : k \\ge n}$.\nBy the [[Hahn–Banach separation theorem|6GWMGFGFJJ]], a norm-closed convex set is weakly closed, so $C_n$ is weakly closed.\nSince $x_k \\to x$ weakly and $x_k \\in C_n$ for $k \\ge n$, the limit $x$ lies in $C_n$.\nHence some convex combination $y_n$ of finitely many $x_k$ with $k \\ge n$ satisfies $\\norm{y_n - x} \u003c 1/n$.\n\nIn the sequence space $\\ell^2$, the unit vectors $e_n$ converge weakly to $0$ but $\\norm{e_n} = 1$ for all $n$.\nThe averages $y_n = \\frac{1}{n}(e_n + e_{n+1} + \\cdots + e_{2n-1})$ have $\\norm{y_n} = 1/\\sqrt{n}$, so they tend to $0$ in norm, as the lemma predicts.\n\n## Explanations\n\n### L1 human\n\nWeak convergence is a loose kind of convergence, and a weakly convergent sequence may stay far from its limit.\nThe lemma says that averaging repairs this.\nSuitable averages of the later terms of the sequence do converge in the usual sense of distance.\n\n### L2 human\n\nThe lemma is a quantitative form of the fact that a convex set has the same closure in the norm topology and in the weak topology.\nIt is used in the calculus of variations and in partial differential equations: one obtains a weak limit from compactness.\nThen one uses the lemma to pass to strong limits of averages, which interact well with convex functionals.\nFor example, it shows that a convex lower semicontinuous functional on a Banach space is weakly lower semicontinuous.\n",
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