Nuclear Space - Properties

Properties

Nuclear spaces are in many ways similar to finite-dimensional spaces and have many of their good properties.

• A closed bounded subset of a nuclear Frechet space is compact. (A bounded subset B of a topological vector space is one such that for any neighborhood U of 0 we can find a positive real scalar λ such that B is contained in λU.) This statement may be paraphrased as a Heine–Borel theorem for nuclear Frechet spaces, analogous to the finite-dimensional situation.

• Any subspace and any quotient space by a closed subspace of a nuclear space is nuclear.

• If A is nuclear and B is any locally convex topological vector space, then the natural map from the projective tensor product of A and B 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 A.

• 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 Frechet 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).