{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T02:39:14Z","timestamp":1777516754009,"version":"3.51.4"},"reference-count":12,"publisher":"SAGE Publications","issue":"4","license":[{"start":{"date-parts":[[2016,9,6]],"date-time":"2016-09-06T00:00:00Z","timestamp":1473120000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Computability"],"published-print":{"date-parts":[[2017,10,31]]},"abstract":"<jats:p>Suppose p is a computable real so that [Formula: see text]. It is shown that the halting set can compute a surjective linear isometry between any two computable copies of [Formula: see text]. It is also shown that this result is optimal in that when [Formula: see text] there are two computable copies of [Formula: see text] with the property that any oracle that computes a linear isometry of one onto the other must also compute the halting set. Thus, [Formula: see text] is [Formula: see text]-categorical and is computably categorical if and only if [Formula: see text]. It is also demonstrated that there is a computably categorical Banach space that is not a Hilbert space. These results hold in both the real and complex case.<\/jats:p>","DOI":"10.3233\/com-160065","type":"journal-article","created":{"date-parts":[[2016,9,6]],"date-time":"2016-09-06T11:15:33Z","timestamp":1473160533000},"page":"391-408","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":16,"title":["Computable copies of \u2113p"],"prefix":"10.1177","volume":"6","author":[{"given":"Timothy H.","family":"McNicholl","sequence":"first","affiliation":[{"name":"Department of Mathematics, Iowa State University, Ames, Iowa 50011, USA."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2016,9,6]]},"reference":[{"key":"ref001","unstructured":"C.J.\u00a0Ash and J.\u00a0Knight, Computable Structures and the Hyperarithmetical Hierarchy, Studies in Logic and the Foundations of Mathematics, Vol.\u00a0144, North-Holland Publishing Co., Amsterdam, 2000."},{"key":"ref002","unstructured":"S.\u00a0Banach, Theory of Linear Operations, North-Holland Mathematical Library, Vol.\u00a038, North-Holland Publishing Co., Amsterdam, 1987, Translated from the French by F. 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