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Of course, $X$ can be orbital only if the algebraic dimension of $X$ is finite or infinite countable. We characterize orbital and strongly orbital metrizable locally convex spaces. We also show that every countably dimensional metrizable locally convex space $X$ does not have the invariant subset prop"},"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":"1209.0973","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-09-05T13:37:07Z","cross_cats_sorted":[],"title_canon_sha256":"2c9d5903b8844ce59fad4dde696a0b0acf36244c827df9d80753491e8c9a94df","abstract_canon_sha256":"fe150c42e51df013e23e88e0e2d19324f8f84b9abc30521bd4d7fe451fcd8a87"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:15.781764Z","signature_b64":"aCl7v3sah4PRDwGWRL9ZmjjCd/hIR0vK1UjhJ7bbHR2le/skfV7JPD9+KJ/xfTndzVsmApmYVCOO1q8nTKA0CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f77c81c1837c9893e114c09ccaad04c2e095f9e7360f9e6f6958b884415e5ed3","last_reissued_at":"2026-05-18T03:46:15.781017Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:15.781017Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Orbital and strongly orbital spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Stanislav Shkarin","submitted_at":"2012-09-05T13:37:07Z","abstract_excerpt":"We say that a (countably dimensional) topological vector space $X$ is orbital if there is $T\\in L(X)$ and a vector $x\\in X$ such that $X$ is the linear span of the orbit ${T^nx:n=0,1,...}$. 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