Ядролық кеңістіктер: Еуропалық кеңістіктердің жалпыламасы

A generalization of finite dimensional Euclidean spaces different from Hilbert spaces
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 not a Banach space, then there is a good chance that it is nuclear.

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

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. Suppose that and are locally convex space with is nuclear. If is nuclear then the vector space of continuous linear maps endowed with the topology of simple convergence is a nuclear space. If is a semi reflexive space whose strong dual is nuclear and if is nuclear then the vector space of continuous linear maps (endowed with the topology of uniform convergence on bounded subsets of ) is a nuclear space.

If is a set of any cardinality, then and (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 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. The space of smooth functions on any compact manifold is nuclear. The Schwartz space of smooth functions on for which the derivatives of all orders are rapidly decreasing is a nuclear space. The space of entire holomorphic functions on the complex plane is nuclear. The space of distributions the strong dual of is nuclear.

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). If is a quasi complete (i. e. all closed and bounded subsets are complete) nuclear space then 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. If is nuclear and is any locally convex topological vector space, then the natural map from the projective tensor product of A and to the injective tensor product is an isomorphism. Roughly speaking this means that there is only one sensible way to define the tensor product. This property characterizes nuclear spaces
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).

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.