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Nuclear space


In mathematics, a nuclear space is a topological vector space with many of the good properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold.

All finite-dimensional vector spaces are nuclear (because every operator on a finite-dimensional vector space is nuclear). There are no Banach spaces that are nuclear, except for the finite-dimensional ones. In practice a sort of converse to this is often true: if a "naturally occurring" topological vector space is not a Banach space, then there is a good chance that it is nuclear.

Much of the theory of nuclear spaces was developed by Alexander Grothendieck while investigating the Schwartz kernel theorem and published in (Grothendieck 1955).

This section lists some of the more common definitions of a nuclear space. The definitions below are all equivalent. Note that some authors use a more restrictive definition of a nuclear space, by adding the condition that the space should be Fréchet. (This means that the space is complete and the topology is given by a countable family of seminorms.)

We start by recalling some background. A locally convex topological vector space V has a topology that is defined by some family of seminorms. For any seminorm, the unit ball is a closed convex symmetric neighborhood of 0, and conversely any closed convex symmetric neighborhood of 0 is the unit ball of some seminorm. (For complex vector spaces, the condition "symmetric" should be replaced by "balanced".) If p is a seminorm on V, we write Vp for the Banach space given by completing V using the seminorm p. There is a natural map from V to Vp (not necessarily injective).


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