The topology on nuclear spaces can be defined by a family of seminorms whose unit balls decrease rapidly in size.
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.
Much of the theory of nuclear spaces was developed by Alexander Grothendieck while investigating the Schwartz kernel theorem and published in (Grothendieck 1955).
has the topology of uniform convergence on bounded subsets) and furthermore, both of these spaces are canonically TVS-isomorphic to
[1] In short, the Schwartz kernel theorem states that:
Another set of motivating examples comes directly from geometry and smooth manifold theory[3]appendix 2.
(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.
be a locally convex topological vector space.
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).
A locally convex topological vector space
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".)
The condition of being a nuclear operator is subtle, and more details are available in the corresponding article.
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 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.
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
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
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
In fact it is sufficient to check this just for Banach spaces
Much of the theory of nuclear spaces was developed by Alexander Grothendieck while investigating the Schwartz kernel theorem and published in (Grothendieck 1955).
the Bochner–Minlos theorem (after Salomon Bochner and Robert Adol'fovich Minlos) guarantees the existence and uniqueness of a corresponding probability measure
, thereby extending the inverse Fourier transform to nuclear spaces.
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.