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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1504.04202","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-04-16T12:13:39Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"5c90190ba836ed5a4594e54cb85942b1bbe013a755813bc9783639220efaf4dd","abstract_canon_sha256":"fada4a3303316883103ec785df7cb71b6975d2c6a0d5f072d7c7ddf95b997d65"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:39.505493Z","signature_b64":"4PWHy5/uPegjyZJSvOi0xZnvvr54mpyP1hmjr9ueQ8yiujMC3LrASxlnUjKBkBdfMU/QHWe+0akMq52FbUc/CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf1e462c1bb878f7033b047444f61aa397b62761dc37f40e169df1aea84c6030","last_reissued_at":"2026-05-18T02:18:39.504735Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:39.504735Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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