Nuclear Space - Examples

Examples

• Any finite-dimensional vector space is nuclear, because any 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 not a Banach space, then there is a good chance that it is nuclear.

• The simplest infinite example of a nuclear space is the space of all rapidly decreasing sequences c=(c₁, c₂,...). ("Rapidly decreasing" means that c_np(n) is bounded for any polynomial p.) For each real number s, we can define a norm ||·||_s by

||c||_s = sup |c_n|ns

If the completion in this norm is C_s, then there is a natural map from C_s to C_t whenever s≥t, and this is nuclear whenever s>t+1, essentially because the series Σnt−s is then absolutely convergent. In particular for each norm ||·||_t we can find another norm, say ||·||_t+2, such that the map from C_t+2 to C_t is nuclear. So the space is nuclear.

• The space of smooth functions on any compact manifold is nuclear.

• The Schwartz space of smooth functions on for which the derivatives of all orders are rapidly decreasing is a nuclear space.

• The space of entire holomorphic functions on the complex plane is nuclear.

• The inductive limit of a sequence of nuclear spaces is nuclear.

• The strong dual of a nuclear Frechet space is nuclear.

• The product of a family of nuclear spaces is nuclear.

• The completion of a nuclear space is nuclear (and in fact a space is nuclear if and only if its completion is nuclear).

• The tensor product of two nuclear spaces is nuclear.