The purpose of this note is to give an example of a distinguished Fréchet space and a non-distinguished Fréchet space which have the same inductive dual. Accordingly, distinguishedness is a property which is not reflected in the inductive dual. In contrast to this example, it was known that the properties of being quasinormable or having the density condition can be characterized in terms of the inductive dual of a Fréchet space.

It is proved, using so-called multirectangular invariants, that the condition αβ = α̃β̃ is sufficient for the isomorphism of the spaces $E₀\left(exp\alpha i\right)\otimes ̂E_{\infty }\left(exp\beta j\right)$ and $E₀\left(exp\alpha ̃i\right)\otimes ̂E_{\infty }\left(exp\beta ̃j\right)$. This solves a problem posed in [14, 15, 1]. Notice that the necessity has been proved earlier in [14].

Let Λ_R(α) be a nuclear power series space of finite or infinite type with lim_{j→∞} (1/j) log α_j = 0. We consider open polydiscs D_a in Λ_R(α)'_b with finite radii and the spaces H(D_a) of all holomorphic functions on D_a under the compact-open topology. We characterize all isomorphy classes of the spaces {H(D_a) | a ∈ Λ_R(α), a > 0}. In the case of a nuclear power series space Λ₁(α) of finite type we give this characterization in terms of the invariants (Ω̅ ) and (Ω̃ ) known from the theory...