Nuclear space

In mathematics, a nuclear space is a topological vector space with many of the good properties of finite-dimensional vector spaces. The topology on them 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 (because every operator on a finite-dimensional vector space is 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 formula_1 and formula_2 the canonical map formula_3 is an isomorphism of TVSs (where formula_4 has the topology of uniform convergence on bounded subsets) and furthermore, both of these spaces are canonically TVS-isomorphic to formula_5 (where since formula_6 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 formula_8 with formula_9 (which is a reflexive space that is even isomorphic to its own strong dual space) and replaces formula_10 with the dual of this formula_9 space. 
Why does such a nice result hold for the space of distributions and test functions but not for the Hilbert space formula_9 (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 theoryappendix 2. Given smooth manifolds formula_13 and a locally convex Hausdorff topological vector space, then there are the following isomorphisms of nuclear spaces
Using standard tensor products for formula_16 as a vector space, the functionformula_17cannot be expressed as a function formula_18 for formula_19 This gives an example demonstrating there is a strict inclusion of setsformula_20
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.
Definition 0: Let formula_21 be a locally convex topological vector space. Then formula_21 is nuclear if for any locally convex space formula_23 the canonical vector space embedding formula_24 is an embedding of TVSs whose image is dense in the codomain (where the domain formula_25 is the projective tensor product and the codomain is the space of all separately continuous bilinear forms on formula_26 endowed with the topology of uniform convergence on equicontinuous subsets). 
We start by recalling some background. A locally convex topological vector space formula_21 has a topology that is defined by some family of seminorms. For any seminorm, the unit ball is a closed convex symmetric neighborhood of the origin, and conversely any 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 formula_28 is a seminorm on formula_29 then formula_30 denotes the Banach space given by completing the auxiliary normed space using the seminorm formula_31 There is a natural map formula_32 (not necessarily injective).
If formula_33 is another seminorm, larger than formula_28 (pointwise as a function on formula_21), then there is a natural map from formula_36 to formula_30 such that the first map factors as formula_38 These maps are always continuous. The space formula_21 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 any seminorm formula_28 we can find a larger seminorm formula_33 so that the natural map formula_42 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 any neighborhood of 0 contains a "much smaller" neighborhood. It is not necessary to check this condition for all seminorms formula_28; 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. 
We will say that a seminorm formula_28 is a Hilbert seminorm if formula_30 is a Hilbert space, or equivalently if formula_28 comes from a sesquilinear positive semidefinite form on formula_47
Definition 2: A nuclear space is a topological vector space with a topology defined by a family of Hilbert seminorms, such that for any Hilbert seminorm formula_28 we can find a larger Hilbert seminorm formula_33 so that the natural map from formula_36 to formula_30 is trace class.
Some authors prefer to use Hilbert–Schmidt operators rather than trace class operators. This makes little difference, because any 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 any Hilbert seminorm formula_28 we can find a larger Hilbert seminorm formula_33 so that the natural map from formula_36 to formula_30 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 any seminorm formula_28 the natural map from formula_32 is nuclear.
Definition 5: A nuclear space is a locally convex topological vector space such that any 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 formula_58 such that for any locally convex topological vector space formula_59 the natural map from the projective to the injective tensor product of formula_58 and formula_59 is an isomorphism.
In fact it is sufficient to check this just for Banach spaces formula_62 or even just for the single Banach space formula_63 of absolutely convergent series.
Characterizations.
Let formula_21 be a Hausdorff locally convex space. Then the following are equivalent:
If formula_21 is a Fréchet space then the following are equivalent:
Sufficient conditions.
Suppose that formula_99 and formula_100 are locally convex space with formula_100 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 formula_21 is nuclear, formula_135 is locally convex, and formula_136 is a continuous bilinear form on formula_137 Then formula_136 originates from a space of the form formula_139 where formula_140 and formula_141 are suitable equicontinuous subsets of formula_92 and formula_143 Equivalently, formula_136 is of the form, 
where formula_147 and each of formula_148 and formula_149 are equicontinuous. Furthermore, these sequences can be taken to be null sequences (i.e. converging to 0) in formula_150 and formula_151 respectively.
Bochner–Minlos theorem.
A continuous functional formula_152 on a nuclear space formula_58 is called a characteristic functional if formula_154 and for any complex formula_155 and formula_156 formula_157 
Given a characteristic functional on a nuclear space formula_159 the Bochner–Minlos theorem (after Salomon Bochner and Robert Adol'fovich Minlos) guarantees the existence and uniqueness of the corresponding probability measure formula_160 on the dual space formula_161 given by
This extends the inverse Fourier transform to nuclear spaces.
In particular, if formula_58 is the nuclear space
where formula_165 are Hilbert spaces, the Bochner–Minlos theorem guarantees the existence of a probability measure with the characteristic function formula_166 that is, the existence of the Gaussian measure on the dual space. Such measure is called white noise measure. When formula_58 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 formula_28 there exists a larger seminorm formula_33 so that the natural map formula_42 is a strongly nuclear.