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Approximation of vector valued smooth functions

A real locally convex space is said to be convenient if it is separated, bornological and Mackey‐complete. These spaces serve as underlying objects for a whole theory of differentiation and integration (see [4]) upon which infinite dimensional differential geometry is based (cf. [8]). We investigate the question of density of the subspaces C∞(E) ⊗ F and 𝒫f (E) ⊗ F of smooth (polynomial) decomposable functions in the space C∞(E, F) of smooth functions between convenient vector spaces E, F with respect to various natural structures. A characterization is given for density with respect to the c∞‐topology and also some classical locally convex topologies on C∞(E, F). It is shown furthermore, that for the space ℝ(ℕ) the convenient analogon of the Schwartz kernel theorem for C∞‐functions holds. Spaces of C∞‐functions on both separable and non‐separable manifolds are considered and an example of a non‐separable manifold is given failing the above property of approximability by decomposable functions. Those notions and features of the theory of convenient vector spaces which are essential for the results of this paper are explained in the introductory section below and where needed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)