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  "body": "\n## Summary\n\nThe bidual $X^{**}$ of a normed space $X$ is the dual of its [[continuous dual space|4A82S5FVRT]] $X^*$.\nThe canonical embedding $J \\colon X \\to X^{**}$ sends a vector $x$ to the functional \"evaluate at $x$\".\nBy the [[Hahn–Banach theorem|QB5AE3VYZZ]], $J$ is a linear isometry, so every normed space sits inside its bidual.\nSpaces for which $J$ is onto are the [[reflexive spaces|NN8RWPC8WG]].\n\n## Explanations\n\n### L1 human\n\nA vector $x$ can be turned into a machine that eats a functional $f$ and returns the number $f(x)$.\nThis machine is linear and bounded, so it is an element of the bidual.\nIn finite dimensions every element of the bidual arises this way, which is the familiar fact that a space is naturally the dual of its dual.\nIn infinite dimensions the bidual can be bigger: the bidual of the space $c_0$ of null sequences is the space $\\ell^\\infty$ of bounded sequences.\n",
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