---
tag: XC6XPMC2FQ
kind: concept
name: Nuclear space
msc: [46A11]
specializes: [B6GY2KJMWW]
sources:
  - url: https://en.wikipedia.org/wiki/Nuclear_space
same_as:
  wikidata: Q283804
  wikipedia: Nuclear space
---

## Summary

A nuclear space is a [[locally convex space|B6GY2KJMWW]] whose seminorms can be compared through [[nuclear operators|XPANM7C1D3]].
These are maps given by absolutely summable series of rank-one maps.
Nuclear spaces were introduced by Alexander Grothendieck in his work on topological tensor products.
They behave in many ways like finite-dimensional spaces: tensor products are well behaved, and the [[Schwartz kernel theorem|2ZKWZ0A4MT]] holds for them.
The Schwartz space and spaces of smooth functions on open sets of $\R^n$ are nuclear, while no infinite-dimensional normed space is.

## Explanations

### L1 human

In a nuclear space, passing from one seminorm to a stronger one makes things very much smaller, in a summable way.
For smooth functions, a bound on many derivatives forces the function to be very close to a finite combination of simple functions.
This is why nuclear spaces feel almost finite-dimensional.
For example, in the Schwartz space of rapidly decreasing smooth functions, bounded sets are relatively compact, as in $\R^n$.
Infinite-dimensional Banach spaces are never nuclear.