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  "body": "\n## Summary\n\nA nuclear space is a [[locally convex space|B6GY2KJMWW]] whose seminorms can be compared through [[nuclear operators|XPANM7C1D3]].\nThese are maps given by absolutely summable series of rank-one maps.\nNuclear spaces were introduced by Alexander Grothendieck in his work on topological tensor products.\nThey behave in many ways like finite-dimensional spaces: tensor products are well behaved, and the [[Schwartz kernel theorem|2ZKWZ0A4MT]] holds for them.\nThe Schwartz space and spaces of smooth functions on open sets of $\\R^n$ are nuclear, while no infinite-dimensional normed space is.\n\n## Explanations\n\n### L1 human\n\nIn a nuclear space, passing from one seminorm to a stronger one makes things very much smaller, in a summable way.\nFor smooth functions, a bound on many derivatives forces the function to be very close to a finite combination of simple functions.\nThis is why nuclear spaces feel almost finite-dimensional.\nFor example, in the Schwartz space of rapidly decreasing smooth functions, bounded sets are relatively compact, as in $\\R^n$.\nInfinite-dimensional Banach spaces are never nuclear.\n",
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