Nuclear space - Imported Knowlpedia Content

Definition

Let $E$ be a Hausdorff locally convex space. For a continuous seminorm $p$ , let $E_p$ be the Banach completion of $E/\ker p$ . The space $E$ is nuclear if, for every continuous seminorm $p$ , there is a continuous seminorm $q\geq p$ such that the canonical map

E_q\longrightarrow E_p

is a nuclear operator: it has an absolutely summable rank-one decomposition. This is a property of the locally convex topology, not merely of the underlying vector space or its topological dual as a set.

Why the definition is strong

The transition map condition forces finite-dimensional-like summability between successively stronger seminorms. It implies that the projective and injective locally convex tensor-product constructions with $E$ agree after the appropriate Hausdorff completion. This removes a major ambiguity in kernel theorems and makes continuous multilinear maps unusually tractable Trèves, Chapters 50–51.

Examples and nonexamples

Finite-dimensional locally convex spaces are nuclear. The Schwartz space $\mathcal S(\mathbb R^n)$ , the test-function space $C_c^\infty(\Omega)$ , and the space $C^\infty(M)$ on a compact smooth manifold are fundamental infinite-dimensional examples with their standard locally convex topologies. By contrast, a Banach space is nuclear as a locally convex space only when it is finite-dimensional: applying the definition to a norm would make an identity-type transition map nuclear, hence compact.

Topological cautions

Nuclearity does not mean that every element is a rapidly decreasing function, nor does it define a class of distributions. Those are properties of particular function spaces and their continuous duals. Changing the topology on a fixed vector space can change whether it is nuclear, because continuity of the seminorms and nuclearity of the completion maps both depend on that topology Schaefer–Wolff, Chapter IV.

References

François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967. Publisher record. Relevant: Chapters 50–51 on nuclear spaces and kernel theorems.
Helmut H. Schaefer and Manfred P. Wolff, Topological Vector Spaces, 2nd ed., Springer, 1999. Publisher record. Relevant: Chapter IV on nuclear maps and spaces.