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  "body": "\n## Definition\n\nLet $X$ be a [[normed vector space|03J5RAKB30]] over $\\R$ or $\\C$, and let $J \\colon X \\to X^{**}$ be the [[canonical embedding into the bidual|QKR3D65TTH]], $(Jx)(f) = f(x)$.\nThe space $X$ is *reflexive* if $J$ is surjective.\n\n## Remarks\n\nSince $J$ is a linear isometry, $X$ is reflexive exactly when $J$ is an isometric isomorphism of $X$ onto $X^{**}$.\nA reflexive space is complete, because $X^{**}$ is complete.\nExample: for $1 \u003c p \u003c \\infty$ and $1/p + 1/q = 1$, the duals satisfy $(\\ell^p)^* = \\ell^q$ and $(\\ell^q)^* = \\ell^p$; under these identifications $J$ is the identity of $\\ell^p$, so $\\ell^p$ is reflexive.\nEvery [[Hilbert space|HM2HHE3J51]] is reflexive by the [[Riesz representation theorem|TTC40N49M6]].\nNon-example: $c_0$ is not reflexive, since $J$ is the inclusion $c_0 \\subset \\ell^\\infty$, which misses the constant sequence $(1, 1, \\ldots)$.\nThe spaces $\\ell^1$, $\\ell^\\infty$ and $C[0, 1]$ are not reflexive either.\nThe definition asks for the specific map $J$ to be onto: the [[James space|DYAWGXE2GV]] is isometrically isomorphic to its bidual by some other map, but it is not reflexive.\n",
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