Some Remarks on Diametral Dimension and Approximate Diametral Dimension of Certain Nuclear Fréchet Spaces

Abstract

The diametral dimension, ∆(E), and the approximate diametral dimension, δ(E), of a nuclear Fréchet space E which satisfies DN and Ω, is related to power series spaces Λ 1 (ε) and Λ ∞ (ε) for some exponent sequence ε. In this article, we examine a question of whether δ(E) must coincide with that of a power series space if ∆(E) does the same, and vice versa.In this regard, we first show that this question has an affirmative answer in an infinite type case by proving the fact that ∆(E) = ∆ (Λ ∞ (ε)) if and only if δ(E) = δ(Λ ∞ (ε)). Then we consider the question in the finite type case and, among other things, we prove that δ(E) = δ (Λ 1 (ε)) if and only if ∆(E) = ∆(Λ 1 (ε)) and E has a prominent bounded subset.