In
mathematics, nuclear spaces are
topological vector spaces
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
that can be viewed as a generalization of finite dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
s and share many of their desirable properties. Nuclear spaces are however quite different from
Hilbert spaces
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
, 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
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
s 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
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example is ...
. All finite-dimensional vector spaces are nuclear. There are no
Banach spaces
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
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
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).
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 In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an i ...
that is even isomorphic to its own strong dual space) and replaces
with the dual of this
space.
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
appendix 2. Given smooth manifolds
and a locally convex Hausdorff topological vector space, then there are the following isomorphisms of nuclear spaces
*
*
Using standard tensor products for
as a vector space, the function
cannot be expressed as a function
for
This gives an example demonstrating there is a strict inclusion of sets
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
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
. (This means that the space is complete and the topology is given by a family of seminorms.)
The following definition was used by Grothendieck to define nuclear spaces.
Definition 0: Let
be a locally convex topological vector space. Then
is nuclear if for any 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
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
has a topology that is defined by some family of
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
s. 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
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ci ...
".)
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 any 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 any 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
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
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 In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace ...
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 any Hilbert seminorm
we can find a larger Hilbert seminorm
so that the natural map from
to
is
trace class In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace ...
.
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
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 any seminorm
the natural map from
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
such that for any 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:
#
is nuclear;
# for any locally convex space
the canonical vector space embedding
is an embedding of TVSs whose image is dense in the codomain;
# for any
Banach space the canonical vector space embedding
is a surjective isomorphism of TVSs;
# for any locally convex Hausdorff space
the canonical vector space embedding
is a surjective isomorphism of TVSs;
# the canonical embedding of