# Nuclear space

A locally convex space for which all continuous linear mappings into an arbitrary Banach space are nuclear operators (cf. Nuclear operator). The concept of a nuclear space arose [1] in an investigation of the question: For what spaces are the analogues of Schwartz' kernel theorem valid (see Nuclear bilinear form)? The fundamental results in the theory of nuclear spaces are due to A. Grothendieck [1]. The function spaces used in analysis are, as a rule, Banach or nuclear spaces. Nuclear spaces play an important role in the spectral analysis of operators on Hilbert spaces (the construction of rigged Hilbert spaces, expansions in terms of generalized eigen vectors, etc.) (see [2]). Nuclear spaces are closely connected with measure theory on locally convex spaces (see [3]). Nuclear spaces can be characterized in terms of dimension-type invariants (approximative dimension, diametral dimension, etc.) (see [2], [4], [5]). One of these invariants is the functional dimension, which for many spaces consisting of entire analytic functions is the same as the number of variables on which these functions depend (see [2]).

In their properties, nuclear spaces are close to finite-dimensional spaces. Every bounded set in a nuclear space is pre-compact. If a nuclear space is complete (or at least quasi-complete, that is, every closed bounded set is complete), then it is semi-reflexive (that is, the space coincides with its second dual as a set of elements), and every closed bounded set in it is compact. If a quasi-complete nuclear space is a barrelled space, then it is also a Montel space (in particular, a reflexive space); any weakly-convergent countable sequence in this space converges also in the original topology. A normed space is nuclear if and only if it is finite dimensional. Every nuclear space has the approximation property: Any continuous linear operator in such a space can be approximated in the operator topology of pre-compact convergence by operators of finite rank (that is, continuous linear operators with finite-dimensional ranges). Nevertheless, there are nuclear Fréchet spaces (cf. Fréchet space) that do not have the bounded approximation property; in such a space the identity operator is not the limit of a countable sequence of operators of finite rank in the strong or weak operator topology [6]. Nuclear Fréchet spaces without a Schauder basis have been constructed, and they can have arbitrarily small diametral dimension, that is, they can be arbitrarily near (in a certain sense) to finite-dimensional spaces [7]. For nuclear spaces a counterexample to the problem of invariant subspaces has been constructed: In a certain nuclear Fréchet space one can find a continuous linear operator without non-trivial invariant closed subspaces [8].

### Examples of nuclear spaces.

1) Let be the space of all (real or complex) infinitely-differentiable functions on equipped with the topology of uniform convergence of all derivatives on compact subsets of . The space dual to consists of all generalized functions (cf. Generalized function) with compact support. Let and be the linear subspaces of consisting, respectively, of functions with compact support and of functions that, together with all their derivatives, decrease faster than any power of as . The duals and of and , relative to the standard topology, consist of all generalized functions and of all generalized functions of slow growth, respectively. The spaces , , , , , and , equipped with the strong topology, are complete reflexive nuclear spaces.

2) let be an infinite matrix, where and , . The space of sequences for which for all , with the topology defined by the semi-norms (cf. Semi-norm), is called a Köthe space, and is denoted by . This space is nuclear if and only if for any one can find a such that .

## Heredity properties.

A locally convex space is nuclear if and only if its completion is nuclear. Every subspace (separable quotient space) of a nuclear space is nuclear. The direct sum, the inductive limit of a countable family of nuclear spaces, and also the product and the projective limit of any family of nuclear spaces, is again nuclear.

Let be an arbitrary locally convex space, and let denote its dual equipped with the strong topology. If is nuclear, then is called conuclear. If is arbitrary and is a nuclear space, then the space of continuous linear operators from into is nuclear with respect to the strong operator topology (simple convergence); if is semi-reflexive and conuclear, then is nuclear also in the topology of bounded convergence.

## Metric and dually-metric nuclear spaces.

A locally convex space is called dually metric, or a space of type , if it has a countable fundamental system of bounded sets and if every (strongly) bounded countable union of equicontinuous subsets in is equicontinuous (cf. Equicontinuity). Any strong dual of a metrizable locally convex space is dually metric; the converse is not true. If is a space of type , then is of type (a Fréchet space, that is, complete and metrizable). Examples of nuclear spaces of type are Köthe spaces, and also and ; accordingly, and are nuclear spaces of type . The spaces and are neither metric nor dually metric.

Metric and dually-metric nuclear spaces are separable, and if complete, they are reflexive. The transition to the dual space establishes a one-to-one correspondence between nuclear spaces of type and complete nuclear spaces of type . If is a complete nuclear space of type and if is a nuclear space of type , then , equipped with the topology of bounded convergence, is nuclear and conuclear.

Every nuclear space of type is isomorphic to a subspace of the space of infinitely-differentiable functions on the real line, that is, is a universal space for the nuclear spaces of type (see [10]). A Fréchet space is nuclear if and only if every unconditionally-convergent series (cf. Unconditional convergence) in is absolutely convergent (that is, with respect to any continuous semi-norm). Spaces of holomorphic functions on nuclear spaces of types and have been studied intensively (see [11]).

## Tensor products of nuclear spaces, and spaces of vector functions.

The algebraic tensor product of two locally convex spaces and can be equipped with the projective and injective topologies, and then becomes a topological tensor product. The projective topology is the strongest locally convex topology in which the canonical bilinear mapping is continuous. The injective topology (or the topology of (bi) equicontinuous convergence) is induced by the natural imbedding , where is the dual of equipped with the Mackey topology , and is the space of continuous linear mappings equipped with the topology of uniform convergence on equicontinuous sets in . Under this imbedding goes into the operator , where denotes the value of the functional at . The completion of in the projective (respectively, injective) topology is denoted by (respectively, ).

For to be a nuclear space it is necessary and sufficient that for any locally convex space the projective and injective topologies in coincide, that is,

(1) |

Actually, it suffices to require that (1) holds for , the space of summable sequences, or for equal to a fixed space with an unconditional basis (see [12]). Nevertheless, there is a (non-nuclear) infinite-dimensional separable Banach space such that (see [13]). If and are complete spaces and is nuclear, then the imbedding can be extended to an isomorphism between and .

If is a non-null nuclear space, then is nuclear if and only if is nuclear. If and are both spaces of type (or ) and if is nuclear, then .

Let be a complete nuclear space consisting of scalar functions (not all) on a certain set ; let also be the inductive limit (locally convex hull) of a countable sequence of spaces of type , and let the topology on be not weaker than the topology of pointwise convergence of functions on . Then for any complete space one can identify with the space of all mappings (vector functions) for which the scalar function belongs to for all . In particular, coincides with the space of all infinitely-differentiable vector functions on with values in , and .

## The structure of nuclear spaces.

Let be a convex circled (i.e. convex balanced) neighbourhood of zero in a locally convex space , and let be the Minkowski functional (continuous semi-norm) corresponding to . Let be the quotient space with the norm induced by , and let be the completion of the normed space . There is defined a continuous canonical linear mapping ; if contains a neighbourhood , then the continuous linear mapping is defined canonically.

For a locally convex space the following conditions are equivalent: 1) is nuclear; 2) has a basis of convex circled neighbourhoods of zero such that for any the canonical mapping is a nuclear operator; 3) the mapping is nuclear for any convex circled neighbourhood of zero in ; and 4) every convex circled neighbourhood of zero in contains another such neighbourhood of zero, , such that the canonical mapping is nuclear.

Let be a nuclear space. For any neighbourhood of zero in and for any such that there is a convex circled neighbourhood for which is (norm) isomorphic to a subspace of the space of sequences with summable -th powers. Thus, coincides with a subspace of the projective limit of a family of spaces isomorphic to . In particular (the case ), in any nuclear space there is a basis of neighbourhoods of zero such that all the spaces are Hilbert spaces; thus, is Hilbertian, that is, the topology in can be generated by a family of semi-norms each of which is obtained from a certain non-negative definite Hermitian form on . Any complete nuclear space is isomorphic to the projective limit of a family of Hilbert spaces. A space of type is nuclear if and only if it can be represented as the projective limit of a countable family of Hilbert spaces , such that the are nuclear operators (or, at least, Hilbert–Schmidt operators, cf. Hilbert–Schmidt operator) for .

## Bases in nuclear spaces.

In a nuclear space every equicontinuous basis is absolute. In a space of type any countable basis (even if weak) is an equicontinuous Schauder basis (cf. Basis), so that in a nuclear space of type any basis is absolute (in particular, unconditional). A similar result holds for complete nuclear spaces of type , and for all nuclear spaces for which the closed-graph theorem holds. A quotient space of a nuclear space of type with a basis does not necessarily have a basis (see [4], [5], [6]).

Let be a nuclear space of type . A topology can be defined in by a countable system of semi-norms , where for all . If has a basis or a continuous norm, then the semi-norms can be taken as norms. Let be a basis in ; then any can be expressed as an (absolutely and unconditionally) convergent series

where the coordinates have the form , and the functionals form a bi-orthogonal basis in . is isomorphic to the Köthe space , where ; under this isomorphism goes into the sequence of its coordinates. A basis in is equivalent to the basis (that is, it can be obtained from by an isomorphism) if and only if and coincide as sets [4]. A basis is called regular (or proper) if there is a system of norms and a permutation of indices such that is monotone decreasing for all . If a nuclear space of type has a regular basis, then any two bases in are quasi-equivalent (that is, they can be made equivalent by a permutation and a normalization of the elements of one of them). There are other sufficient conditions for all bases in to be quasi-equivalent (see [4], [14]). A complete description of the class of nuclear spaces with this property is not known (1984).

Example. The Hermite functions form a basis in the complete metric nuclear space of smooth functions on the real line that are rapidly decreasing together with all their derivatives. is isomorphic to .

#### References

[1] | A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc. (1955) |

[2] | I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1968) (Translated from Russian) |

[3] | R.A. Minlos, "Generalized random processes and their extension in measure" Trudy Moskov. Mat. Obshch. , 8 (1959) pp. 497–518 (In Russian) |

[4] | B.S. Mityagin, "Approximate dimension and bases in nuclear spaces" Russian Math. Surveys , 16 : 4 pp. 59–127 Uspekhi Mat. Nauk , 16 : 4 (1961) pp. 63–132 |

[5] | A. Pietsch, "Nuclear locally convex spaces" , Springer (1972) (Translated from German) |

[6] | E. Dubinsky, "Structure of nuclear Fréchet spaces" , Springer (1979) |

[7] | N.M. Zobin, B.S. Mityagin, "Examples of nuclear linear metric spaces without a basis" Funct. Anal. Appl. , 8 : 4 (1974) pp. 304–313 Funktsional. Anal. i Prilozhen. , 8 : 4 (1974) pp. 35–47 |

[8] | A. Atzmon, "An operator without invariant subspaces on a nuclear Fréchet space" Ann. of Math. , 117 : 3 (1983) pp. 669–694 |

[9] | H.H. Schaefer, "Topological vector spaces" , Springer (1971) |

[10] | T. Komura, Y. Komura, "Ueber die Einbettung der nuklearen Räume in " Math. Ann. , 162 (1965–1966) pp. 284–288 |

[11] | S. Dineen, "Complex analysis in locally convex spaces" , North-Holland (1981) |

[12] | K. John, V. Zizler, "On a tensor product characterization of nuclearity" Math. Ann. , 244 : 1 (1979) pp. 83–87 |

[13] | G. Pisier, "Contre-example à une conjecture de Grothendieck" C.R. Acad. Sci. Paris , 293 (1981) pp. 681–683 (English abstract) |

[14] | M.M. Dragilev, "Bases in Köthe spaces" , Rostov-on-Don (1983) (In Russian) |

#### Comments

A generalized function is also called a distribution, and a generalized function of slow growth is also called a tempered distribution.

Let be a topological linear space, a neighbourhood of zero in , a set in , and a (small) positive number. An -set for relative to a neighbourhood of zero is a set such that for every there is a such that . Let be the smallest number of elements in -sets for relative to . The functional dimension of is defined by

where range over the neighbourhoods of zero in . Cf. [2], Sect. I.3.8 for more details.

Let be a locally convex space and consider two neighbourhoods of zero such that absorbs , i.e. for some positive number . Let

This number is called the -th diameter of with respect to . The diametral dimension of a locally convex space is the collection of all sequences of non-negative numbers with the property that for each neighbourhood of zero there is a neighbourhood of zero absorbed by for which , .

A locally convex space is nuclear if and only if for some (respectively, each) positive number the sequence belongs to the diametral dimension of . See [5], Chapt. 9 for more details.

Let again be neighbourhoods of zero of a locally convex space such that absorbs . The -content of with respect to is the supremum of all natural numbers such that there are with for all . The approximative dimension of a locally convex space is the collection of all positive functions on such that for each neighbourhood of zero there is a neighbourhood of zero absorbed by such that

The number is defined by the exponential rate of growth of as . More precisely,

A locally convex space is nuclear if and only if for some (respectively, each) positive number the following condition is satisfied: For each neighbourhood of zero there is a neighbourhood of zero absorbed by such that . Cf. [5], Chapt. 9 for more details.

Let be a bounded circled neighbourhood of a topological vector space . The Minkowski functional associated to is defined by

This is well-defined for each since is absorbent (i.e. for each there is an such that ). Cf. [a7], Sects. 15.10, 16.4.

#### References

[a1] | A. Grothendieck, "Résumé de la théorie métrique des produits tensoriels topologiques" Bol. Soc. Mat. Sao-Paulo , 8 (1956) pp. 1–79 |

[a2] | A. Grothendieck, "Topological vector spaces" , Gordon & Breach (1973) (Translated from French) |

[a3] | H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German) |

[a4] | G. Pisier, "Factorization of linear operators and geometry of Banach spaces" , Amer. Math. Soc. (1986) |

[a5] | G. Pisier, "Counterexamples to a conjecture of Grothendieck" Acta. Math. , 151 (1983) pp. 181–208 |

[a6] | J.F. Colombeau, "Differential calculus and holomorphy" , North-Holland (1982) |

[a7] | G. Köthe, "Topological vector spaces" , 1 , Springer (1969) |

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Nuclear space.

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