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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $E$ be a Hausdorff [[locally convex space|B6GY2KJMWW]] over $\\mathbb{K}$.\nFor a continuous [[seminorm|FP5GVCN8GV]] $p$ on $E$, let $E_p$ be the Banach space obtained by completing the quotient $E / \\set{x : p(x) = 0}$ with the norm induced by $p$.\nIf $q \\ge p$ are continuous seminorms, the identity of $E$ induces a continuous linear map $\\iota_{q, p} \\colon E_q \\to E_p$ of norm at most $1$.\n$E$ is *nuclear* if the following holds.\nFor every continuous seminorm $p$ on $E$ there is a continuous seminorm $q \\ge p$ such that $\\iota_{q, p}$ is a [[nuclear operator|XPANM7C1D3]].\n\n## Remarks\n\nExample: the Schwartz space $\\mathcal{S}(\\R^n)$, the space $C^\\infty(\\Omega)$ of smooth functions on an open set $\\Omega \\subseteq \\R^n$, and the space $H(\\Omega)$ of holomorphic functions on an open set $\\Omega \\subseteq \\C$ are nuclear.\nNon-example: an infinite-dimensional normed space $E$ is not nuclear.\nTake $p$ to be the norm; then every continuous seminorm $q \\ge p$ is equivalent to the norm.\nSo $\\iota_{q, p}$ is an isomorphism of infinite-dimensional Banach spaces, which is not compact and hence not nuclear.\nGrothendieck's original definition is equivalent: $E$ is nuclear if and only if, for every Banach space $F$, the projective and injective tensor topologies on $E \\otimes F$ agree.\nEvery nuclear Fréchet space is a Montel space.\n",
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