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  "body": "\n## Statement\n\nLet $X$ be a real [[Banach space|06GZ956657]] with [[continuous dual space|4A82S5FVRT]] $X^*$, and let $C \\subseteq X$ be a nonempty, closed, bounded [[convex set|02AS45T3H9]].\nSay that $f \\in X^*$ attains its supremum on $C$ when $f(c) = \\sup_{x \\in C} f(x)$ for some $c \\in C$.\nThen the set of such functionals is [[dense|0BSJSX7ME2]] in $X^*$ for the norm topology.\nIn particular, taking $C$ to be the closed unit ball, the functionals $f \\in X^*$ with $f(x) = \\norm{f}$ for some $x$ with $\\norm{x} \\le 1$ are dense in $X^*$.\n\n## Remarks\n\nProof idea: given $f \\in X^*$ and $\\varepsilon \u003e 0$, one considers a cone determined by $f$ and $\\varepsilon$.\nA maximality argument, in the spirit of Zorn's lemma or of Ekeland's principle, gives $c \\in C$ such that the cone translated to $c$ meets $C$ only at $c$.\nThe Hahn–Banach separation theorem then gives a functional $g$ that attains its supremum on $C$ at $c$ and satisfies $\\norm{f - g} \\le \\varepsilon$.\n\nExample: let $X = c_0$ be the space of real sequences tending to $0$, with the supremum norm.\nThe functional $f(x) = \\sum_{n \\ge 1} 2^{-n} x_n$ does not attain its norm $1$ on the unit ball, since that would need $x_n = 1$ for all $n$.\nThe truncations $g_N(x) = \\sum_{n \\le N} 2^{-n} x_n$ attain their norm at $x = (1, \\ldots, 1, 0, 0, \\ldots)$ and converge to $f$ in norm, as the theorem predicts.\n\nThe density cannot be improved to equality in general: by James' theorem, every functional attains its norm on the unit ball only when $X$ is reflexive.\nThe analogue for bounded operators between Banach spaces, in place of functionals, fails in general.\n\n## Explanations\n\n### L1 human\n\nA functional attains its norm if some vector of length at most $1$ realizes its largest value.\nIn infinite dimensions this can fail, because the unit ball is not compact.\nThe theorem says that such failures are rare: every functional can be approximated as closely as one likes by functionals that do attain their norm.\n\n### L2 human\n\nNorm-attaining functionals correspond to supporting hyperplanes of the unit ball, so the theorem says that support points of closed bounded convex sets are plentiful.\nIt is used in the geometry of Banach spaces, for instance in the study of smoothness and of subdifferentials of convex functions.\nThe complex case follows from the real one, by viewing a complex Banach space as a real one.\n",
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