The Lie Algebra of a Nuclear Group

The Lie algebra of a nuclear group is a locally convex nuclear vector space. Mathematics Subject Classification (2000): 22B05, 22E65, 46Axx 1. The Lie Algebra of Abelian Topological Groups Let G be an abelian topological group. The set of all one–parameter–subgroups L(G) := {λ : R→ G : λ is a continuous homomorphism} is called the Lie algebra of G . We define addition pointwise and scalar multiplication by the following formula R× L(G)→ L(G), (s, λ) 7→ (s · λ : t 7→ λ(st)). It is topologized with the compact–open topology. We use the notation P (K,U) := {f : X → Y : f(K) ⊆ U} for sets X, Y and subsets K ⊆ X and U ⊆ Y . If no confusion can arise, we will use the same notation P (·, ·) for sets of continuous functions, respectively, continuous homomorphisms. The system of neighbourhoods of the neutral element 0 of an abelian topological group G is denoted by UG(0). We set G∗ := {χ : G → T : χ is a continuous homomorphism} where T is the compact group of complex numbers of modulus 1. With multiplication defined pointwise and endowed with the compact–open topology, G∗ is an abelian topological group, named dual group of G . We introduce the canonical homomorphism αG : G→ G∗∗, x 7→ (χ 7→ χ(x)). The famous Pontryagin van–Kampen duality theorem states that if G is a locally compact abelian (LCA) group then αG is a topological isomorphism. [A proof can be found in [3], p. 351 or [4], p.378 or [7], p.84.] Lemma 1.1. αG is continuous if and only if every compact subset of G ∗ is equicontinuous. If G is metrizable (more generally, a k–space) then αG is continuous. ISSN 0949–5932 / $2.50 c © Heldermann Verlag