For nuclear Frechet spaces with DN and Omega, equality of the diametral dimension with an infinite-type power series space is equivalent to equality of the approximate diametral dimension; for finite-type spaces the equivalence holds when the space has a prominent bounded set.
Bessaga, Some remarks on Dragilev’s theorem , Studia Math
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Some Remarks on Diametral Dimension and Approximate Diametral Dimension of Certain Nuclear Fr\'echet Spaces
For nuclear Frechet spaces with DN and Omega, equality of the diametral dimension with an infinite-type power series space is equivalent to equality of the approximate diametral dimension; for finite-type spaces the equivalence holds when the space has a prominent bounded set.