{"paper":{"title":"The Ascoli property for function spaces and the weak topology of Banach and Fr\\'echet spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.FA","authors_text":"G. Plebanek, J. Kakol, S. Gabriyelyan","submitted_at":"2015-04-16T12:13:39Z","abstract_excerpt":"Following [3] we say that a Tychonoff space $X$ is an Ascoli space if every compact subset $\\mathcal{K}$ of $C_k(X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_\\mathbb{R}$-space, hence any $k$-space, is Ascoli.\n  Let $X$ be a metrizable space. We prove that the space $C_{k}(X)$ is Ascoli iff $C_{k}(X)$ is a $k_\\mathbb{R}$-space iff $X$ is locally compact. Moreover, $C_{k}(X)$ endowed with the weak topology is Ascoli iff $X$ is countable and discrete.\n  Using some basic concepts from probability theory and measure-theoretic properties of $\\ell"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04202","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}