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  "body": "\n## Statement\n\nLet $X$ be a [[Banach space|0BDTTM24DP]] over $\\R$ or $\\C$, and let $X'$ be the space of continuous linear functionals on $X$.\nLet $(x_n)_{n \\ge 1}$ be a sequence in $X$.\nAssume that for every strictly increasing sequence $n_1 \u003c n_2 \u003c \\cdots$ of indices there is $y \\in X$ with\n$$\\varphi(y) = \\sum_{k=1}^{\\infty} \\varphi(x_{n_k}) \\quad \\text{for all } \\varphi \\in X'.$$\nThen for every such sequence of indices the series $\\sum_{k} x_{n_k}$ converges in norm.\nIn particular, the series $\\sum_n x_n$ is [[unconditionally convergent|2ME9M61TJK]] in norm.\n\n## Remarks\n\nThe hypothesis says that every subseries converges in the weak topology; the conclusion says that every subseries converges in norm.\nThe theorem needs all subseries: in $\\ell^2$, the series $e_1 + (e_2 - e_1) + (e_3 - e_2) + \\cdots$ has partial sums $e_n$, which tend to $0$ weakly but not in norm.\nThe theorem also holds in Hausdorff locally convex spaces, with the weak topology given by the continuous dual.\n\n## Explanations\n\n### L1 human\n\nIn an infinite-dimensional space, weak convergence is in general much weaker than convergence in norm.\nFor series, the theorem says that this gap closes as soon as one asks for every subseries.\nIf each subseries converges when tested by every continuous linear functional, then each subseries already converges in norm.\n\n### L2 human\n\nThe theorem is a basic tool for [[series in Banach spaces|WN6M8B3ZKB]] and for vector measures.\nA countably additive vector measure is defined by convergence of $\\sum \\mu(A_n)$ for disjoint sets $A_n$, and every subfamily of disjoint sets gives a subseries.\nSo the theorem shows that a weakly countably additive vector measure is countably additive in norm.\nIt is also used in the Bochner and Pettis theories of integration of vector-valued functions.\n",
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