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  "body": "\n## Definition\n\nLet $X$ be a [[normed vector space|03J5RAKB30]] over $\\mathbb{K} = \\R$ or $\\C$, and let $X^*$ be its [[continuous dual space|4A82S5FVRT]] with the operator norm.\nThe *bidual* of $X$ is the Banach space $X^{**} = (X^*)^*$.\nThe *canonical embedding* is the map $J \\colon X \\to X^{**}$ given by\n$$(Jx)(f) = f(x) \\quad \\text{for } x \\in X \\text{ and } f \\in X^*.$$\n\n## Remarks\n\nFor each $x$, the map $Jx$ is linear on $X^*$ and $\\abs{(Jx)(f)} \\le \\norm{f}\\,\\norm{x}$, so $Jx \\in X^{**}$ with $\\norm{Jx} \\le \\norm{x}$.\nThe [[Hahn–Banach theorem|QB5AE3VYZZ]] gives $f$ with $\\norm{f} = 1$ and $f(x) = \\norm{x}$, so in fact $J$ is a [[linear isometry|07SJXTWVQW]].\nExample: for $X = \\mathbb{K}^n$ with any norm, $J$ is an isometric isomorphism onto $X^{**}$, because $\\dim X^{**} = \\dim X$; compare [[the double dual in linear algebra|0ZJM9D7NPS]].\nExample: for $X = c_0$ one has $X^* = \\ell^1$ and $X^{**} = \\ell^\\infty$, and $J$ is the inclusion of $c_0$ in $\\ell^\\infty$, which is not onto.\nNon-example: the algebraic double dual, the space of all linear forms on $X^*$, is in general larger than $X^{**}$, because it does not require continuity.\nThe image $J(X)$ is closed in $X^{**}$ exactly when $X$ is complete; in general the closure of $J(X)$ is a [[completion|2W7R5JHF0C]] of $X$.\n",
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