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  "body": "\n## Summary\n\nLet $X$ and $Y$ be Banach spaces.\nTheir projective tensor product is the completion of the algebraic [[tensor product|0BF2DE2SYK]] $X \\otimes Y$ for the largest reasonable norm on it.\nIt turns bounded bilinear maps on $X \\times Y$ into bounded linear maps, with the same norm.\nIts dual is the space of bounded linear maps from $X$ to the dual of $Y$.\nWith the [[injective tensor product|RQXC5D3M1Y]] it is one of the two extreme ways to make $X \\otimes Y$ a Banach space, as studied by Grothendieck.\n\n## Explanations\n\n### L1 human\n\nA tensor $u$ in $X \\otimes Y$ can be written in many ways as a sum of simple products $x_i \\otimes y_i$.\nEach way has a cost, the sum of the products of the lengths $\\norm{x_i} \\, \\norm{y_i}$.\nThe projective norm of $u$ is the cheapest cost over all ways to write it.\nFor example, in $\\ell^1 \\otimes \\ell^1$ this norm turns a tensor into a double sequence $(a_{mn})$ with norm $\\sum_{m,n} \\abs{a_{mn}}$.\n",
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