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.

has the topology of uniform convergence on bounded subsets is nuclear, this tensor product is simultaneously the injective tensor product and projective tensor product). where all of these TVS-isomorphisms are canonical.

(which is a reflexive space space. The question of why such a nice result holds for the space of distributions and test functions but not for the Hilbert space (which is generally considered one of the "nicest" TVSs) 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 theoryappendix 2 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 is the projective tensor product endowed with the topology of uniform convergence on equicontinuous subsets).

We start by recalling some background. A locally convex topological vector space 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 denotes the Banach space given by completing the auxiliary normed space (not necessarily injective).

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 nuclear.

; 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 is a Hilbert seminorm

Definition 2: A nuclear space 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 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 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 an isomorphism.

of absolutely convergent series.

Characterizations

be a Hausdorff locally convex space. Then the following are equivalent:

1. is nuclear;

2. is an embedding of TVSs whose image is dense in the codomain;

3. for any Banach space is a surjective isomorphism of TVSs;

4. is a surjective isomorphism of TVSs;

5. is a surjective isomorphism of TVSs;

6. is a surjective TVS-isomorphism.

7. is nuclear;

8. is nuclear;

9. is trace class;

10. is Hilbert–Schmidt;

11. is nuclear.

12. any continuous linear map to a Banach space is nuclear;

13. is prenuclear;

14. every equicontinuous is prenuclear;

15. that transforms the unit ball into an equicontinuous set, is nuclear;

16. is a nuclear space;

is a Fréchet space then the following are equivalent:

1. is nuclear;

2. every unconditionally summable sequence is absolutely summable;

3. is nuclear;

Sufficient conditions

• A locally convex Hausdorff space is nuclear if and only if its completion is nuclear.

• Every subspace of a nuclear space is nuclear.

• Every Hausdorff quotient space of a nuclear space is nuclear.

• The inductive limit of a countable sequence of nuclear spaces is nuclear.

• The locally convex direct sum of a countable sequence of nuclear spaces is nuclear.

The strong dual of a nuclear Fréchet space is nuclear.

• In general, the strong dual of a nuclear space may fail to be nuclear.

• A Fréchet space whose strong dual is nuclear is itself nuclear.

• The limit of a family of nuclear spaces is nuclear.

• The product of a family of nuclear spaces is nuclear.

• The completion of a nuclear space is nuclear (and in fact a space is nuclear if and only if its completion is nuclear).

• The tensor product of two nuclear spaces is nuclear.

• The projective tensor product, as well as its completion, of two nuclear spaces is nuclear.

is nuclear.

• endowed with the topology of simple convergence is a nuclear space.

• is a semi-reflexive ) is a nuclear space.

Examples

(with the product topology) are both nuclear spaces.

is nuclear. So the space is nuclear.

• The space of smooth functions on any compact manifold is nuclear.

• The space of entire holomorphic functions on the complex plane is nuclear.

• Both the Schwartz space for which the derivatives of all orders are rapidly decreasing, and its strong dual space, the space of tempered distrutions, are nuclear.

• and its strong dual, the space of distributions are nuclear.

Properties

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

• Every finite-dimensional Hausdorff space is nuclear.

• A Fréchet space is nuclear if and only if its strong dual is nuclear.

• Every bounded subset of a nuclear space is precompact (recall that a set is precompact if its closure in the completion of the space is compact). This is analogous to the Heine-Borel theorem. In contrast, no infinite-dimensional normed space has this property (although the finite-dimensional spaces do).

• is a quasi-complete has the Heine-Borel property.

• A nuclear quasi-complete barrelled space is a Montel space.

• Every closed equicontinuous subset of the dual of a nuclear space is a compact metrizable set (for the strong dual topology).

• Every nuclear space is a subspace of a product of Hilbert spaces.

• Every nuclear space admits a basis of seminorms consisting of Hilbert norms.

• Every nuclear space is a Schwartz space.

• Every nuclear space possesses the approximation property.

• Any subspace and any quotient space by a closed subspace of a nuclear space is nuclear.

• is any locally convex topological vector space, then the natural map from the projective tensor product of A

• In the theory of measures on topological vector spaces, a basic theorem states that any continuous cylinder set measure on the dual of a nuclear Fréchet space automatically extends to a Radon measure. This is useful because it is often easy to construct cylinder set measures on topological vector spaces, but these are not good enough for most applications unless they are Radon measures (for example, they are not even countably additive in general).

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: respectively.

Bochner–Minlos theorem

Any continuous positive-definite functional is called a characteristic functional

the Bochner–Minlos theorem (after Salomon Bochner and Robert Adol'fovich Minlos) guarantees the existence and uniqueness of a corresponding probability measure

is the Fourier-Stieltjes transform, thereby extending the inverse Fourier transform to nuclear spaces.

that is, the existence of the Gaussian measure on the dual space. Such measure is called white noise measure is the Schwartz space, the corresponding random element is a random distribution.

Strongly nuclear spaces

A strongly nuclear space is strongly nuclear.

See also

• Auxiliary normed space

• Fredholm kernel – type of a kernel on a Banach space

• Injective tensor product

• Locally convex topological vector space – a vector space with a topology defined by convex open sets

• Nuclear operator

• Projective tensor product – a tensor product defined on two topological vector spaces

• Rigged Hilbert space – a construction linking the study of "bound" and continuous eigenvalues in functional analysis

• Trace class – the class of compact operators for which a finite trace can be defined

• Topological vector space – a vector space with a notion of nearness

Notes

References

• Takeyuki Hida and Si Si, Lectures on white noise functionals, World Scientific Publishing, 2008.