Vol. 92, No. 1, 1981

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Holomorphy on spaces of distribution

Philip J. Boland and Sean Dineen

Vol. 92 (1981), No. 1, 27–34

If E is a locally convex space and U E is open, then H(U) is the space of holomorphic functions on U (i.e., H(U) = {f : U C, f G-analytic and continuous}). τ0 is the topology of uniform convergence on compact subsets of U. τω is the Nachbin ported topology defined by all semi-norms on H(U) ported by compact subsets of U. (A semi-norm p on H(U) is ported by K if whenever V is open and K V U, there exists CV such that p(f) CV |f|V for all f H(U).) τδ is the topology defined by all semi-norms p on H(U) with the following property: if (Un) is a countable increasing open cover of U, there exist C > 0 and UN such that p(f) C|f|UN for all f H(U). HHY (U) is the space of hypoanalytic functions on U — that is HHY (U) = {f : f is G -analytic and the restriction of f to any compact set  K U is continuous}.

If Ω is open in Rn, then 𝒟(Ω) and 𝒟′(Ω) are respectively the Schwartz space of test functions and the Schwartz space of distributions on Ω. We prove that H(𝒟(Ω))HHY (𝒟(Ω)) and that τ0 = τω = τδ on H(𝒟(Ω)) while HHY (𝒟′(Ω)) = H(𝒟′(Ω)) but τ0τωτδ on H(𝒟′(Ω)).

Mathematical Subject Classification 2000
Primary: 46G20
Secondary: 30H05, 32A07, 46F99
Received: 18 April 1979
Revised: 6 December 1979
Published: 1 January 1981
Philip J. Boland
Sean Dineen