# Some Remarks on Diametral Dimension and Approximate Diametral Dimension of Certain Nuclear Fr\'echet Spaces

## TL;DR

This paper investigates the relationship between diametral and approximate diametral dimensions of nuclear Fréchet spaces, establishing conditions under which these dimensions coincide with those of specific power series spaces.

## Contribution

It proves that in infinite type cases, the diametral and approximate diametral dimensions coincide with those of power series spaces, and explores similar conditions in finite type cases.

## Key findings

- In infinite type cases, $ riangle(E)$ equals that of a power series space if and only if $oldsymbol{ riangle(E)}$ equals the approximate dimension.
- In finite type cases, $oldsymbol{ riangle(E)}$ equals that of a power series space if and only if $oldsymbol{ riangle(E)}$ does, and $E$ has a prominent bounded subset.

## Abstract

The diametral dimension, $\Delta(E)$, and the approximate diametral dimension, $\delta (E)$, of a nuclear Fr\'echet space $E$ which satisfies $\underline{DN}$ and $\Omega$, is related to power series spaces $\Lambda_{1}(\varepsilon)$ and $\Lambda_{\infty}\left(\varepsilon\right)$ for some exponent sequence $\varepsilon$. 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 an infinite type case by proving the fact that $\Delta (E)=\Delta\left(\Lambda_{\infty} (\varepsilon)\right)$ if and only if $\delta (E)= \delta (\Lambda_{\infty}(\varepsilon))$. Then we consider the question in the finite type case and, among other things, we prove that $\delta (E)=\delta\left(\Lambda_{1} (\varepsilon)\right)$ if and only if $\Delta (E)= \Delta (\Lambda_{1}(\varepsilon))$ and $E$ has a prominent bounded subset.