Die metrisierbaren linearen Teilräume des Raumes$\mathfrak{O}_M$ von L. Schwartz

Abstract

In Schwartz' terminology, a real or complex valued functionf, defined and infinitely differentiable on ℝ _n , belongs to$\mathfrak{O}_M$ iff, as well as any of its derivatives, is at most of polynomial growth. The topology of$\mathfrak{O}_M$ is defined by the seminorms sup{∣ϕ(x)D ^p f(x)∣;x∈ℝⁿ}, where ϕ belongs to$\mathfrak{S}$ andD ^p is any derivative. It is well-known that$\mathfrak{O}_M$ is non-metrisable. For any μ: ℕⁿ→ℕ, let$\mathfrak{B}_\mu$ be the space of all infinitely differentiable functionsf satisfying, for eachp, sup{∣(1+∣x∣²)^−μ(p) D ^p f(x)∣;x∈ℝⁿ}<∞, with the obvious topology. These spaces, which are of very little use elsewhere in the theory of distributions, can be conveniently applied to characterise the metrisable linear subspaces of$\mathfrak{O}_M$: A linear subspace of$\mathfrak{O}_M$ is metrisable if and only if it is, algebraically and topologically, a subspace of some$\mathfrak{B}_\mu$.