{"paper":{"title":"Some Remarks on Diametral Dimension and Approximate Diametral Dimension of Certain Nuclear Fr\\'echet Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Nazl{\\i} Do\\u{g}an","submitted_at":"2019-08-05T20:18:38Z","abstract_excerpt":"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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01838","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.01838/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}