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  "body": "\n## Statement\n\nLet $X$ be a [[Banach space|06GZ956657]] over $\\R$ or $\\C$, with the [[weak topology|P6KHVSTPZV]] $\\sigma(X, X^*)$, and let $A \\subseteq X$.\nThen $A$ is [[compact|0R6EWFKRCF]] in the weak topology if and only if the following holds.\nEvery sequence in $A$ has a subsequence that converges weakly to a point of $A$.\n\n## Remarks\n\nThe weak topology of an infinite-dimensional space is not metrizable, so compactness and sequential compactness need not agree a priori.\nThe theorem says they agree for the weak topology of a Banach space.\nFor \"compact implies sequentially compact\", a sequence lies in a separable closed subspace $Y$.\nOn weakly compact subsets of $Y$ the weak topology is metrizable, because countably many functionals separate the points of $Y$.\nThe converse direction is harder.\nOne proof first shows that $A$ is bounded, then uses the [[Banach–Alaoglu theorem|1BXEJXQ2YT]] in $X^{**}$.\nIt then shows that the weak* closure of the image $J(A)$ under the [[canonical embedding|QKR3D65TTH]] $J$ lies in $J(X)$.\nThe same equivalences hold for relative compactness: the weak closure of $A$ is weakly compact exactly when every sequence in $A$ has a weakly convergent subsequence with limit in $X$.\n\nExample: the closed unit ball of a [[reflexive space|NN8RWPC8WG]] is weakly compact, so every bounded sequence in a reflexive space has a weakly convergent subsequence.\nThe analogue fails for the weak* topology: the unit ball of $(\\ell^\\infty)^*$ is weak*-compact by Banach–Alaoglu, but the evaluations $\\delta_n(x) = x_n$ have no weak* convergent subsequence.\n\n## Explanations\n\n### L1 human\n\nIn general topology, compactness is defined with open covers, while in metric spaces one can test it with sequences.\nThe weak topology of a Banach space is not given by a metric, yet the theorem says that sequences still suffice to test compactness.\nThis makes weak compactness much easier to handle in practice.\n\n### L2 human\n\nThe theorem is used in analysis and partial differential equations whenever one extracts weakly convergent subsequences from bounded sequences in reflexive spaces such as $L^p$ for $1 \u003c p \u003c \\infty$ or Sobolev spaces.\nIt also combines with [[James's theorem|G1VWNQSAK5]] and the [[Krein–Smulian theorem|NYVHMQJP3E]] in the theory of weakly compact sets.\n",
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