Nuclear space

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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.

Contents

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 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 ( Grothendieck 1955 ). 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). [1] 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 (which is a reflexive space that is even isomorphic to its own strong dual space) and replaces with the dual of this space. [2] Why does such a nice result hold for the space of distributions and test functions but not for the Hilbert space (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 [3] appendix 2. Given smooth manifolds 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 countable family of seminorms.)

The following definition was used by Grothendieck to define nuclear spaces. [4]

Definition 0: Let be a locally convex topological vector space. Then is nuclear if for every locally convex space 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 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 is a seminorm on then denotes the Banach space given by completing the auxiliary normed space using the seminorm There is a natural map (not necessarily injective).

If is another seminorm, larger than (pointwise as a function on ), then there is a natural map from to such that the first map factors as These maps are always continuous. The space 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 we can find a larger seminorm so that the natural map 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 ; 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 is a Hilbert seminorm if is a Hilbert space, or equivalently if comes from a sesquilinear positive semidefinite form on

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 we can find a larger Hilbert seminorm so that the natural map from to 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 we can find a larger Hilbert seminorm so that the natural map from to 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 the natural map from 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 such that for every locally convex topological vector space the natural map from the projective to the injective tensor product of and is an isomorphism.

In fact it is sufficient to check this just for Banach spaces or even just for the single Banach space of absolutely convergent series.

Characterizations

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

  1. is nuclear;
  2. for any locally convex space the canonical vector space embedding is an embedding of TVSs whose image is dense in the codomain;
  3. for any Banach space the canonical vector space embedding is a surjective isomorphism of TVSs; [5]
  4. for any locally convex Hausdorff space the canonical vector space embedding is a surjective isomorphism of TVSs; [5]
  5. the canonical embedding of in is a surjective isomorphism of TVSs; [6]
  6. the canonical map of is a surjective TVS-isomorphism. [6]
  7. for any seminorm we can find a larger seminorm so that the natural map is nuclear;
  8. for any seminorm we can find a larger seminorm so that the canonical injection is nuclear; [5]
  9. the topology of is defined by a family of Hilbert seminorms, such that for any Hilbert seminorm we can find a larger Hilbert seminorm so that the natural map is trace class;
  10. has a topology defined by a family of Hilbert seminorms, such that for any Hilbert seminorm we can find a larger Hilbert seminorm so that the natural map is Hilbert–Schmidt;
  11. for any seminorm the natural map from is nuclear.
  12. any continuous linear map to a Banach space is nuclear;
  13. every continuous seminorm on is prenuclear; [7]
  14. every equicontinuous subset of is prenuclear; [7]
  15. every linear map from a Banach space into that transforms the unit ball into an equicontinuous set, is nuclear; [5]
  16. the completion of is a nuclear space;

If is a Fréchet space then the following are equivalent:

  1. is nuclear;
  2. every summable sequence in is absolutely summable; [6]
  3. the strong dual of is nuclear;

Sufficient conditions

Suppose that and are locally convex space with is nuclear.

Examples

If is a set of any cardinality, then and (with the product topology) are both nuclear spaces. [12]

A relatively simple infinite-dimensional example of a nuclear space is the space of all rapidly decreasing sequences ("Rapidly decreasing" means that is bounded for any polynomial ). For each real number it is possible to define a norm by

If the completion in this norm is then there is a natural map from whenever and this is nuclear whenever 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 ( Grothendieck 1955 ). We have the following generalization of the theorem.

Schwartz kernel theorem: [9] Suppose that is nuclear, is locally convex, and is a continuous bilinear form on Then originates from a space of the form where and are suitable equicontinuous subsets of and Equivalently, 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 on a nuclear space is called a characteristic functional if and for any and [16] [17]

Given a characteristic functional on a nuclear space the Bochner–Minlos theorem (after Salomon Bochner and Robert Adol'fovich Minlos) guarantees the existence and uniqueness of a corresponding probability measure on the dual space such that

where is the Fourier transform of , thereby extending the inverse Fourier transform to nuclear spaces. [18]

In particular, if 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 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 there exists a larger seminorm so that the natural map is a strongly nuclear.

See also

Related Research Articles

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References

  1. Trèves 2006, p. 531.
  2. Trèves 2006, pp. 509–510.
  3. Costello, Kevin (2011). Renormalization and effective field theory. Providence, R.I.: American Mathematical Society. ISBN   978-0-8218-5288-0. OCLC   692084741.
  4. Schaefer & Wolff 1999, p. 170.
  5. 1 2 3 4 Trèves 2006, p. 511.
  6. 1 2 3 Schaefer & Wolff 1999, p. 184.
  7. 1 2 Schaefer & Wolff 1999, p. 178.
  8. 1 2 3 4 5 6 Schaefer & Wolff 1999, p. 103.
  9. 1 2 3 4 5 Schaefer & Wolff 1999, p. 172.
  10. Schaefer & Wolff 1999, p. 105.
  11. 1 2 Schaefer & Wolff 1999, p. 173.
  12. Schaefer & Wolff 1999, p. 100.
  13. Schaefer & Wolff 1999, p. 101.
  14. Trèves 2006, p. 520.
  15. Schaefer & Wolff 1999, p. 110.
  16. Holden et al. 2009, p. 258.
  17. Simon 2005, pp. 10–11.
  18. T. R. Johansen, The Bochner-Minlos Theorem for nuclear spaces and an abstract white noise space , 2003.

Bibliography