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  "body": "\n## Statement\n\nLet $X$ be a [[Banach space|06GZ956657]] over $\\R$ or $\\C$, and let $X^*$ be its [[continuous dual space|4A82S5FVRT]].\nThen $X$ is [[reflexive|NN8RWPC8WG]] if and only if every $f \\in X^*$ attains its norm on the closed unit ball.\nThat is, for every $f \\in X^*$ there is $x \\in X$ with $\\norm{x} \\le 1$ and $\\abs{f(x)} = \\norm{f}$.\n\n## Remarks\n\nThe direction \"reflexive implies norm attaining\" is short.\nIn a reflexive space the closed unit ball is weakly compact, and $x \\mapsto \\abs{f(x)}$ is weakly continuous, so it attains its maximum $\\norm{f}$.\nThe converse is the deep part, and its proof by James is long and combinatorial.\n\nThe theorem gives a concrete test for non-reflexivity.\nOn $c_0$, the functional $f(x) = \\sum_{n \\ge 1} 2^{-n} x_n$ has norm $\\sum_n 2^{-n} = 1$.\nFor $x \\in c_0$ with $\\norm{x}_\\infty \\le 1$, some coordinate has $\\abs{x_n} \u003c 1$, so $\\abs{f(x)} \u003c 1$.\nSo $f$ does not attain its norm, and $c_0$ is not reflexive.\n\n## Explanations\n\n### L1 human\n\nEvery bounded linear functional has a norm, which is the supremum of its values on the unit ball.\nThe theorem says that this supremum is always a maximum exactly in the reflexive spaces.\nSo reflexivity, which is defined through the double dual, can be seen by looking at single functionals.\n\n### L2 human\n\nJames's theorem is used in optimization and convex analysis, where the existence of maximizers of linear functionals on the unit ball matters.\nIt contrasts with the Bishop–Phelps theorem, which says that in every Banach space the norm-attaining functionals are dense in $X^*$.\nIt sits beside the [[Eberlein–Šmulian theorem|2MTN8CQWME]] and the [[Goldstine theorem|K4S9G3AQBM]] as a main characterization of weak compactness and reflexivity.\n",
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