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We prove that the strong dual $E'_\\beta$ of a Montel strict $(LF)$-space $E$ is an Ascoli space iff one of the following assertions holds: (i) $E$ is a Fr\\'{e}chet--Montel space, so $E'_\\beta$ is a sequential non-Fr\\'{e}chet--Urysohn space, or (ii) $E=\\phi$, so $E'_\\beta= \\mathbb{R}^\\omega$. Consequently, the space $\\mathcal{D}(\\Omega)$ of test"},"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":"1702.07867","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-02-25T10:00:39Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"1d55fc8912801bc33cc7ec4146e3c01388b20ce26ee66da5a1425bc4219ef705","abstract_canon_sha256":"cb44334a4cff61efd8cfd67dc97f8b4ebde80ef617741621aade881a2d2f02e9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:56.810951Z","signature_b64":"K3X4MAS1ImUsJE7/P5EZ4qYy/yxeWYNEqp3vipEWcfOB1wJpXSsSexEVXSIHX7J5QwApXtQEIWaMZDHPpYndAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f383ab72ba949f98e815abab5e280a03e2ce00d305d17ef1efd7379641c1f58d","last_reissued_at":"2026-05-18T00:49:56.810338Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:56.810338Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Topological properties of strict $(LF)$-spaces and strong duals of Montel strict $(LF)$-spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.FA","authors_text":"Saak Gabriyelyan","submitted_at":"2017-02-25T10:00:39Z","abstract_excerpt":"Following [2], a Tychonoff space $X$ is Ascoli if every compact subset of $C_k(X)$ is equicontinuous. By the classical Ascoli theorem every $k$-space is Ascoli. We show that a strict $(LF)$-space $E$ is Ascoli iff $E$ is a Fr\\'{e}chet space or $E=\\phi$. We prove that the strong dual $E'_\\beta$ of a Montel strict $(LF)$-space $E$ is an Ascoli space iff one of the following assertions holds: (i) $E$ is a Fr\\'{e}chet--Montel space, so $E'_\\beta$ is a sequential non-Fr\\'{e}chet--Urysohn space, or (ii) $E=\\phi$, so $E'_\\beta= \\mathbb{R}^\\omega$. 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