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  "body": "\n## Definition\n\nLet $X$ be a real [[Banach space|06GZ956657]] and let $J \\colon X \\to (-\\infty, +\\infty]$ be a [[functional|1EQM0DJ4NC]].\nA sequence $(u_k)$ in $X$ converges *weakly* to $u$ if $\\ell(u_k) \\to \\ell(u)$ for every continuous linear functional $\\ell$ on $X$.\nThe functional $J$ is *sequentially weakly lower semicontinuous* if\n$$J(u) \\le \\liminf_{k \\to \\infty} J(u_k)$$\nfor every $u \\in X$ and every sequence $(u_k)$ that converges weakly to $u$.\n\n## Remarks\n\nReplacing sequences by the [[weak topology|P6KHVSTPZV]] itself gives weak lower [[semicontinuity|RN4G8SGP6P]] in the topological sense, which implies the sequential form.\nIn the calculus of variations the sequential form is the one that is used.\n\nExample: the norm $J(u) = \\norm{u}$ is weakly lower semicontinuous.\nBy the Hahn–Banach theorem $\\norm{u} = \\sup \\abs{\\ell(u)}$ over continuous linear $\\ell$ with $\\norm{\\ell} \\le 1$.\nFor each such $\\ell$, $\\abs{\\ell(u)} = \\lim_k \\abs{\\ell(u_k)} \\le \\liminf_k \\norm{u_k}$.\n\nNon-example: on $X = L^2(0, 2\\pi)$ let $J(u) = -\\norm{u}^2$ and $u_k(x) = \\sin(kx)$.\nThen $u_k \\to 0$ weakly, by the Riemann–Lebesgue lemma, and $J(u_k) = -\\pi$ for all $k$.\nBut $J(0) = 0 \u003e -\\pi$, so $J$ is not weakly lower semicontinuous, although it is continuous for the norm.\n\nBy Mazur's lemma, a convex functional that is lower semicontinuous for the norm is weakly lower semicontinuous.\n",
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