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  "body": "\n## Summary\n\nLet $X$ and $Y$ be Banach spaces.\nTheir injective tensor product is the completion of the algebraic [[tensor product|0BF2DE2SYK]] $X \\otimes Y$ for the smallest reasonable norm on it.\nThis norm sees a tensor only through its values on pairs of functionals of norm at most $1$.\nIt describes spaces of continuous vector-valued functions: $C(K)$ with the injective tensor product by $X$ is the space of continuous functions from $K$ to $X$.\nWith the [[projective tensor product|685ZV73JJ8]] it is one of the two extreme tensor norms of Grothendieck's theory.\n\n## Explanations\n\n### L1 human\n\nA tensor $u = \\sum_i x_i \\otimes y_i$ gives a number $\\sum_i f(x_i) g(y_i)$ for each pair of functionals $f$ and $g$.\nThe injective norm is the largest such number when $f$ and $g$ have norm at most $1$.\nIf $X$ and $Y$ are spaces of continuous functions on compact spaces $K$ and $L$, the tensor becomes the function $(s, t) \\mapsto \\sum_i x_i(s) y_i(t)$ on $K \\times L$.\nThen the injective norm is just the maximum of its absolute value.\n",
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