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

Doğan N.

BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN, vol.27, no.3, pp.353-368, 2020 (SCI-Expanded) identifier identifier


The diametral dimension, Delta(E), and the approximate diametral dimension, delta(E), of a nuclear Frechet space E which satisfies DN and Omega, are related to corresponding invariant of power series spaces Lambda(1) (epsilon) and Lambda(infinity) (epsilon) for some exponent sequence epsilon. In this article, we examine a question of whether delta(E) must coincide with that of a power series space if Delta(E) does the same, and vice versa. In this regard, we first show that this question has an affirmative answer in the infinite type case by showing that Delta(E) = Delta (Lambda(infinity)(epsilon)) if and only if delta(E) = delta(Lambda(infinity)(epsilon)). Then we consider the question in the finite type case and, among other things, we prove that delta(E) = delta (Lambda(1)(epsilon)) if and only if Delta(E) = Delta(Lambda(1)(epsilon)) and E has a prominent bounded subset.