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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\\mathbb{K} = \\R$ or $\\C$, and let $X \\otimes Y$ be their algebraic [[tensor product|0BF2DE2SYK]].\nFor $u \\in X \\otimes Y$, let\n$$\\pi(u) = \\inf \\Big\\{ \\sum_{i=1}^n \\norm{x_i} \\, \\norm{y_i} : u = \\sum_{i=1}^n x_i \\otimes y_i \\Big\\}.$$\nThe infimum runs over all finite representations of $u$.\nThen $\\pi$ is a norm on $X \\otimes Y$, the *projective norm*.\nThe *projective tensor product* $X \\widehat{\\otimes}_\\pi Y$ is the completion of $X \\otimes Y$ for $\\pi$.\n\n## Remarks\n\nThe norm is a cross norm: $\\pi(x \\otimes y) = \\norm{x} \\, \\norm{y}$.\nThe bound $\\le$ comes from the representation with one term.\nFor $\\ge$, take functionals $f \\in X^*$ and $g \\in Y^*$ of norm $1$ with $f(x) = \\norm{x}$ and $g(y) = \\norm{y}$, given by the Hahn–Banach theorem.\nFor any representation, $\\norm{x} \\, \\norm{y} = \\abs{\\sum_i f(x_i) g(y_i)} \\le \\sum_i \\norm{x_i} \\, \\norm{y_i}$.\n\nThe infimum is needed: $u = 0$ can be written as $x \\otimes y + x \\otimes (-y)$, with cost $2 \\norm{x} \\, \\norm{y}$.\nA single representation does not give a well-defined number.\n\nUniversal property: bounded bilinear maps $B \\colon X \\times Y \\to Z$ into a Banach space $Z$ correspond to bounded linear maps $X \\widehat{\\otimes}_\\pi Y \\to Z$ with the same norm.\nIn particular, the dual of $X \\widehat{\\otimes}_\\pi Y$ is the space of bounded linear maps from $X$ to $Y^*$.\nFor example, $\\ell^1 \\widehat{\\otimes}_\\pi \\ell^1$ is isometric to $\\ell^1(\\N \\times \\N)$.\n",
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