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  "body": "\n## Statement\n\nLet $X$ be a [[normed vector space|03J5RAKB30]] over $\\R$ or $\\C$, with closed unit ball $B_X$.\nLet $J \\colon X \\to X^{**}$ be the [[canonical embedding into the bidual|QKR3D65TTH]].\nLet $B_{X^{**}}$ be the closed unit ball of $X^{**}$.\nThen $J(B_X)$ is [[dense|0BSJSX7ME2]] in $B_{X^{**}}$ for the weak* topology of $X^{**}$, the [[weak topology|P6KHVSTPZV]] defined by the functionals $\\xi \\mapsto \\xi(f)$ for $f \\in X^*$.\n\n## Remarks\n\nThe weak* closure $C$ of $J(B_X)$ is convex and lies in $B_{X^{**}}$, since $B_{X^{**}}$ is weak*-closed.\nIf some $\\xi \\in B_{X^{**}}$ were not in $C$, the [[Hahn–Banach separation theorem|6GWMGFGFJJ]] in $X^{**}$ with the weak* topology would give $f \\in X^*$ with $\\operatorname{Re} \\xi(f) \u003e \\sup_{x \\in B_X} \\operatorname{Re} f(x) = \\norm{f}$.\nThis contradicts $\\norm{\\xi} \\le 1$.\n\nFor $X = c_0$, with $X^{**} = \\ell^\\infty$, the theorem says that every sequence $y$ with $\\sup_n \\abs{y_n} \\le 1$ is a weak* limit of elements of the unit ball of $c_0$.\nIndeed the truncations $(y_1, \\ldots, y_n, 0, 0, \\ldots)$ converge to $y$ against every element of $\\ell^1$.\n\n## Explanations\n\n### L1 human\n\nA normed space may be strictly smaller than its double dual.\nThe theorem says it is still large in a weak sense.\nEvery element of the unit ball of the double dual can be approximated, in the weak* topology, by elements of the unit ball of the space itself.\nSo the double dual adds only limit points of a loose kind.\n\n### L2 human\n\nCombined with the [[Banach–Alaoglu theorem|1BXEJXQ2YT]], the theorem shows that a Banach space is [[reflexive|NN8RWPC8WG]] exactly when its closed unit ball is weakly compact.\nIt is also the key step in the proof of the [[Milman–Pettis theorem|AJ5RG9XQYZ]].\nIn general the approximation needs nets, not sequences; for separable $X^*$ sequences suffice.\n",
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