Nuclear Space - Definition

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 be Fréchet. (This means that the space is complete and the topology is given by a countable family of seminorms.)

We start by recalling some background. A locally convex topological vector space V has a topology that is defined by some family of seminorms. For any seminorm, the unit ball is a closed convex symmetric neighborhood of 0, 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 p is a seminorm on V, we write V_p for the Banach space given by completing V using the seminorm p. There is a natural map from V to V_p (not necessarily injective).

Definition 1: A nuclear space is a locally convex topological vector space such that for any seminorm p we can find a larger seminorm q so that the natural map from V_q to V_p 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 p; 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. (On Hilbert spaces nuclear operators are often called trace class operators.) We will say that a seminorm p is a Hilbert seminorm if V_p is a Hilbert space, or equivalently if p comes from a sesquilinear positive semidefinite form on V.

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 p we can find a larger Hilbert seminorm q so that the natural map from V_q to V_p 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 p we can find a larger Hilbert seminorm q so that the natural map from V_q to V_p 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 p the natural map from V to V_p 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 A such that for any locally convex topological vector space B the natural map from the projective to the injective tensor product of A and B is an isomorphism.

In fact it is sufficient to check this just for Banach spaces B, or even just for the single Banach space l1 of absolutely convergent series.