In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite-dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite-dimensional Euclidean spaces. They were introduced by Alexander Grothendieck. The topology on nuclear spaces 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 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 (where has the topology of uniform convergence on bounded subsets) and furthermore, both of these spaces are canonically TVS-isomorphic to (where since is nuclear, this tensor product is simultaneously the injective tensor product and projective tensor product). 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 C_c^\infty with L^2 (which is a reflexive space that is even isomorphic to its own strong dual space) and replaces with the dual of this L^2 space. Why does such a nice result hold for the space of distributions and test functions but not for the Hilbert space L^2 (which is generally considered one of the "nicest" TVSs)? This question led Grothendieck to discover nuclear spaces, nuclear maps, and the injective tensor product.

Motivations from geometry

Another set of motivating examples comes directly from geometry and smooth manifold theory appendix 2. Given smooth manifolds M,N and a locally convex Hausdorff topological vector space, then there are the following isomorphisms of nuclear spaces

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 also be a Fréchet space. (This means that the space is complete and the topology is given by a family of seminorms.) 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 every locally convex space Y, the canonical vector space embedding is an embedding of TVSs whose image is dense in the codomain (where the domain is the projective tensor product and the codomain is the space of all separately continuous bilinear forms on endowed with the topology of uniform convergence on equicontinuous subsets). We start by recalling some background. A locally convex topological vector space X has a topology that is defined by some family of seminorms. For every seminorm, the unit ball is a closed convex symmetric neighborhood of the origin, and conversely every 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 X, then X_p denotes the Banach space given by completing the auxiliary normed space using the seminorm p. There is a natural map X \to X_p (not necessarily injective). If q is another seminorm, larger than p (pointwise as a function on X), then there is a natural map from X_q to X_p such that the first map factors as These maps are always continuous. The space X 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 every seminorm p we can find a larger seminorm q so that the natural map X_q \to X_p 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 every 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. (On Hilbert spaces nuclear operators are often called trace class operators.) We will say that a seminorm p is a Hilbert seminorm if X_p is a Hilbert space, or equivalently if p comes from a sesquilinear positive semidefinite form on X. Definition 2: A nuclear space is a topological vector space with a topology defined by a family of Hilbert seminorms, such that for every Hilbert seminorm p we can find a larger Hilbert seminorm q so that the natural map from X_q to X_p is trace class. Some authors prefer to use Hilbert–Schmidt operators rather than trace class operators. This makes little difference, because every 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 every Hilbert seminorm p we can find a larger Hilbert seminorm q so that the natural map from X_q to X_p 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 every seminorm p the natural map from X \to X_p is nuclear. Definition 5: A nuclear space is a locally convex topological vector space such that every 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 every 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 \ell^1 of absolutely convergent series.

Characterizations

Let X be a Hausdorff locally convex space. Then the following are equivalent: If X is a Fréchet space then the following are equivalent:

Sufficient conditions

Suppose that X, Y, and N are locally convex space with N is nuclear.

Examples

If d is a set of any cardinality, then \R^d and \Complex^d (with the product topology) are both nuclear spaces. A relatively simple infinite-dimensional example of a nuclear space is the space of all rapidly decreasing sequences ("Rapidly decreasing" means that c_n p(n) is bounded for any polynomial p). For each real number s, it is possible to define a norm by If the completion in this norm is C_s, then there is a natural map from C_s \to C_t whenever s \geq t, and this is nuclear whenever s > t + 1 essentially because the series is then absolutely convergent. In particular for each norm this is possible to find another norm, say such that the map is nuclear. So the space is nuclear.

Properties

Nuclear spaces are in many ways similar to finite-dimensional spaces and have many of their good properties.

The kernel theorem

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 X \times Y. Then v originates from a space of the form where A^{\prime} and B^{\prime} are suitable equicontinuous subsets of X^{\prime} and Y^{\prime}. Equivalently, v is of the form, where and each of and are equicontinuous. Furthermore, these sequences can be taken to be null sequences (that is, convergent to 0) in and respectively.

Bochner–Minlos theorem

Any continuous positive-definite functional C on a nuclear space A is called a characteristic functional if C(0) = 1, and for any x_j \in A and Given a characteristic functional on a nuclear space A, the Bochner–Minlos theorem (after Salomon Bochner and Robert Adol'fovich Minlos) guarantees the existence and uniqueness of a corresponding probability measure \mu on the dual space A^{\prime} such that where C(y) is the Fourier transform of \mu, thereby extending the inverse Fourier transform to nuclear spaces. In particular, if A is the nuclear space where H_k 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 there exists a larger seminorm q so that the natural map X_q \to X_p is a strongly nuclear.