关于某些核Fréchet空间的直径维数和近似直径维数的注记
Some Remarks on Diametral Dimension and Approximate Diametral Dimension of Certain Nuclear Fréchet Spaces
摘要：
满足$\underline{DN}$和$Ω$的核Fréchet空间$E$的直径维数$Δ(E)$和近似直径维数$δ(E)$与幂级数空间$Λ_{1}(\varepsilon)$和$Λ_{\infty}\left(\varepsilon\right)$有关，其中对某些指数序列$\varepsilon$ .在这篇文章中，我们研究的问题是否$δ(E)$必须符合的幂级数空间，如果$Δ(E)$做同样的，反之亦然。在这方面，我们首先表明，这个问题有一个肯定的答案，在一个无限型的情况下，证明了一个事实，即$Δ(E)=Δ\left(Λ_{\infty} (\varepsilon)\right)$当且仅当$δ(E)= δ(Λ_{\infty}(\varepsilon))$。然后，我们考虑的问题，在有限型的情况下，除其他事项外，我们证明了$δ(E)=δ\left(Λ_{1} (\varepsilon)\right)$当且仅当$Δ(E)= Δ(Λ_{1}(\varepsilon))$和$E$有一个突出的有界子集。
英文摘要：
The diametral dimension, $Δ(E)$, and the approximate diametral dimension, $δ(E)$, of a nuclear Fréchet space $E$ which satisfies $\underline{DN}$ and $Ω$, is related to power series spaces $Λ_{1}(\varepsilon)$ and $Λ_{\infty}\left(\varepsilon\right)$ for some exponent sequence $\varepsilon$. In this article, we examine a question of whether $δ(E)$ must coincide with that of a power series space if $Δ(E)$ 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 $Δ(E)=Δ\left(Λ_{\infty} (\varepsilon)\right)$ if and only if $δ(E)= δ(Λ_{\infty}(\varepsilon))$. Then we consider the question in the finite type case and, among other things, we prove that $δ(E)=δ\left(Λ_{1} (\varepsilon)\right)$ if and only if $Δ(E)= Δ(Λ_{1}(\varepsilon))$ and $E$ has a prominent bounded subset.
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