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 : fis G-analytic and the restriction of fto any compact set K ⊂ Uis 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 τ0≠τω≠τδ on H(𝒟′(Ω)).
