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. 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.

Original motivation: the Schwartz kernel theorem

Much of the theory of nuclear spaces was developed by Alexander Grothendieck while investigating the Schwartz kernel theorem and published in. We now describe this motivation.
For any open subsets and, the canonical map is an isomorphism of TVSs and furthermore, both of these spaces are canonically TVS-isomorphic to .
In short, the Schwartz kernel theorem states that:
where all of these TVS-isomorphisms are canonical.
This result is false if one replaces the space with and replaces with the dual of this space.
Why does such a nice result hold for the space of distributions and test functions but not for the Hilbert space ?
This question led Grothendieck to discover nuclear spaces, nuclear maps, and the injective tensor product.

Definition

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.
The following definition was used by Grothendieck to define nuclear spaces.
Definition 0: Let X be a locally convex topological vector space. Then X is nuclear if for any locally convex space Y, the canonical vector space embedding is an embedding of TVSs whose image is dense in the codomain.
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.
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.
If q is another seminorm, larger than p, then there is a natural map from Vq to Vp such that the first map factors as V → Vq → Vp. These maps are always continuous. The space V is nuclear when a stronger condition holds, namely that these maps are nuclear operators. The condition of being a nuclear operator is subtle, and more details are available in the corresponding article.
Definition 1: A nuclear space is a locally convex topological vector space such that for any seminorm p we can find a larger seminorm q so that the natural map from Vq to Vp is nuclear.
Informally, this means that whenever we are given the unit ball of some seminorm, we can find a "much smaller" unit ball of another seminorm inside it, or that any neighborhood of 0 contains a "much smaller" neighborhood. It is not necessary to check this condition for all seminorms p; it is sufficient to check it for a set of seminorms that generate the topology, in other words, a set of seminorms that are a subbase for the topology.
Instead of using arbitrary Banach spaces and nuclear operators, we can give a definition in terms of Hilbert spaces and trace class operators, which are easier to understand.
We will say that a seminorm p is a Hilbert seminorm if Vp is a Hilbert space, or equivalently if p comes from a sesquilinear positive semidefinite form on V.
Definition 2: A nuclear space is a topological vector space with a topology defined by a family of Hilbert seminorms, such that for any Hilbert seminorm p we can find a larger Hilbert seminorm q so that the natural map from Vq to Vp is trace class.
Some authors prefer to use Hilbert–Schmidt operators rather than trace class operators. This makes little difference, because any trace class operator is Hilbert–Schmidt, and the product of two Hilbert–Schmidt operators is of trace class.
Definition 3: A nuclear space is a topological vector space with a topology defined by a family of Hilbert seminorms, such that for any Hilbert seminorm p we can find a larger Hilbert seminorm q so that the natural map from Vq to Vp is Hilbert–Schmidt.
If we are willing to use the concept of a nuclear operator from an arbitrary locally convex topological vector space to a Banach space, we can give shorter definitions as follows:
Definition 4: A nuclear space is a locally convex topological vector space such that for any seminorm p the natural map from V to Vp is nuclear.
Definition 5: A nuclear space is a locally convex topological vector space such that any continuous linear map to a Banach space is nuclear.
Grothendieck used a definition similar to the following one:
Definition 6: A nuclear space is a locally convex topological vector space A such that for any locally convex topological vector space B the natural map from the projective to the injective tensor product of A and B is an isomorphism.
In fact it is sufficient to check this just for Banach spaces B, or even just for the single Banach space l1 of absolutely convergent series.

Characterizations

Let X be a Hausdorff locally convex space. Then the following are equivalent:
  1. X is nuclear;
  2. for any locally convex space Y, the canonical vector space embedding is an embedding of TVSs whose image is dense in the codomain;
  3. for any Banach space Y, the canonical vector space embedding is a surjective isomorphism of TVSs;
  4. for any locally convex Hausdorff space Y, the canonical vector space embedding is a surjective isomorphism of TVSs;
  5. the canonical embedding of in is a surjective isomorphism of TVSs;
  6. the canonical map of is a surjective TVS-isomorphism.
  7. for any seminorm p we can find a larger seminorm q so that the natural map from Vq to Vp is nuclear;
  8. for any seminorm p we can find a larger seminorm q so that the canonical injection is nuclear;
  9. the topology of X is defined by a family of Hilbert seminorms, such that for any Hilbert seminorm p we can find a larger Hilbert seminorm q so that the natural map from Vq to Vp is trace class;
  10. X has a topology defined by a family of Hilbert seminorms, such that for any Hilbert seminorm p we can find a larger Hilbert seminorm q so that the natural map from Vq to Vp is Hilbert–Schmidt;
  11. for any seminorm p the natural map from V to Vp is nuclear.
  12. any continuous linear map to a Banach space is nuclear;
  13. every continuous seminorm on X is prenuclear;
  14. every equicontinuous subset of is prenuclear;
  15. every linear map from a Banach space into that transforms the unit ball into an equicontinuous set, is nuclear;
  16. the completion of X is a nuclear space;
If X is a Fréchet space then the following are equivalent:
  1. X is nuclear;
  2. every summable sequence in X is absolutely summable;
  3. the strong dual of X is nuclear;

    Sufficient conditions

Suppose that X, Y, and N' are locally convex space with N is nuclear.
Nuclear spaces are in many ways similar to finite-dimensional spaces and have many of their good properties.
Much of the theory of nuclear spaces was developed by Alexander Grothendieck while investigating the Schwartz kernel theorem and published in. We have the following generalization of the theorem.
Schwartz kernel theorem: Suppose that X is nuclear, Y is locally convex, and v is a continuous bilinear form on. Then v originates from a space of the form where and are suitable equicontinuous subsets of and. Equivalently, v is of the form,
where and each of and are equicontinuous. Furthermore, these sequences can be taken to be null sequences in and, respectively.

Bochner–Minlos theorem

A continuous functional C on a nuclear space A is called a characteristic functional if C = 1, and for any complex and, j,k = 1,..., n,
Given a characteristic functional on a nuclear space A, the Bochner–Minlos theorem guarantees the existence and uniqueness of the corresponding probability measure on the dual space, given by
This extends the inverse Fourier transform to nuclear spaces.
In particular, if A is the nuclear space
where are Hilbert spaces, the Bochner–Minlos theorem guarantees the existence of a probability measure with the characteristic function, that is, the existence of the Gaussian measure on the dual space. Such measure is called white noise measure. When A is the Schwartz space, the corresponding random element is a random distribution.

Strongly nuclear spaces

A strongly nuclear space is a locally convex topological vector space such that for any seminorm p we can find a larger seminorm q so that the natural map from Vq to Vp is a strongly nuclear.