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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\\mathbb{K} = \\R$ or $\\C$, with [[continuous duals|4A82S5FVRT]] $X^*$ and $Y^*$.\nLet $X \\otimes Y$ be their algebraic [[tensor product|0BF2DE2SYK]].\nFor $u = \\sum_{i=1}^n x_i \\otimes y_i$ in $X \\otimes Y$, let\n$$\\varepsilon(u) = \\sup \\Big\\{ \\Big| \\sum_{i=1}^n f(x_i) \\, g(y_i) \\Big| : f \\in X^*, \\ g \\in Y^*, \\ \\norm{f} \\le 1, \\ \\norm{g} \\le 1 \\Big\\}.$$\nThis number does not depend on the representation of $u$, and $\\varepsilon$ is a norm on $X \\otimes Y$, the *injective norm*.\nThe *injective tensor product* $X \\check{\\otimes}_\\varepsilon Y$ is the completion of $X \\otimes Y$ for $\\varepsilon$.\n\n## Remarks\n\nThe norm is a cross norm: $\\varepsilon(x \\otimes y) = \\norm{x} \\, \\norm{y}$, by the Hahn–Banach theorem.\nIt is at most the [[projective norm|685ZV73JJ8]] $\\pi$.\nThe bounds on $f$ and $g$ are needed: without them, scaling $f$ makes the supremum infinite for every $u \\ne 0$.\n\nExample: let $X = Y = \\R^2$ with the Euclidean norm and $u = e_1 \\otimes e_1 + e_2 \\otimes e_2$.\nThen $\\varepsilon(u) = \\sup \\abs{\\ip{f}{g}} = 1$, over unit vectors $f, g$.\nThe projective norm of $u$ is $2$, so the two norms differ.\n\nFor compact Hausdorff spaces $K$ and $L$, the map $f \\otimes g \\mapsto ((s, t) \\mapsto f(s) g(t))$ extends to an isometry of $C(K) \\check{\\otimes}_\\varepsilon C(L)$ onto $C(K \\times L)$.\nMore generally, $C(K) \\check{\\otimes}_\\varepsilon X$ is the space of continuous functions from $K$ to $X$ with the sup norm.\n",
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