{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T07:11:57Z","timestamp":1648624317066},"reference-count":4,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":5855,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1998,3]]},"abstract":"<jats:p>For <jats:italic>x, y<\/jats:italic> \u03f5 \u211d<jats:sup>\u03c9<\/jats:sup> define the inner product<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200015292_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>which may not be finite or even exist. We say that <jats:italic>x<\/jats:italic> and <jats:italic>y<\/jats:italic> are orthogonal if (<jats:italic>x, y<\/jats:italic>) converges and equals 0.<\/jats:p><jats:p>Define <jats:italic>l<jats:sub>p<\/jats:sub><\/jats:italic> to be the set of all <jats:italic>x<\/jats:italic> \u03f5 \u211d<jats:sup>\u03c9<\/jats:sup> such that<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200015292_eqnU2\" \/><\/jats:disp-formula><\/jats:p><jats:p>For Hilbert space, <jats:italic>l<jats:sub>2<\/jats:sub><\/jats:italic>, any family of pairwise orthogonal sequences must be countable. For a good introduction to Hilbert space, see Retherford [4].<\/jats:p><jats:p>T<jats:sc>heorem<\/jats:sc> 1. <jats:italic>There exists a pairwise orthogonal family F of size continuum such that F is a subset of l<jats:sub>p<\/jats:sub> for every p &gt; 2<\/jats:italic>.<\/jats:p><jats:p>It was already known that there exists a family of continuum many pairwise orthogonal elements of \u211d<jats:sup>\u03c9<\/jats:sup>. A family <jats:italic>F<\/jats:italic> \u2286 \u211d<jats:sup>\u03c9<\/jats:sup>\u2216<jats:bold>0<\/jats:bold> of pairwise orthogonal sequences is orthogonally complete or a maximal orthogonal family iff the only element of \u211d<jats:sup>\u03c9<\/jats:sup> orthogonal to every element of <jats:italic>F<\/jats:italic> is <jats:bold>0<\/jats:bold>, the constant 0 sequence.<\/jats:p><jats:p>It is somewhat surprising that Kunen's perfect set of orthogonal elements is maximal (a fact first asserted by Abian). MAD families, nonprincipal ultrafilters, and many other such maximal objects cannot be even Borel.<\/jats:p><jats:p>T<jats:sc>heorem<\/jats:sc> 2. <jats:italic>There exists a perfect maximal orthogonal family of elements of \u211d<jats:sup>\u03c9<\/jats:sup><\/jats:italic>.<\/jats:p><jats:p>Abian raised the question of what are the possible cardinalities of maximal orthogonal families.<\/jats:p><jats:p>T<jats:sc>heorem<\/jats:sc> 3. <jats:italic>In the Cohen real model there is a maximal orthogonal set in \u211d<jats:sup>\u03c9<\/jats:sup> of cardinality \u03c9<jats:sub>1<\/jats:sub>, but there is no maximal orthogonal set of cardinality \u03ba with \u03c9<jats:sub>1<\/jats:sub> &lt; \u03ba &lt; \u03f2<\/jats:italic>.<\/jats:p><jats:p>By the Cohen real model we mean any model obtained by forcing with finite partial functions from \u03b3 to 2, where the ground model satisfies GCH and \u03b3<jats:sup>\u03c9<\/jats:sup> = \u03b3.<\/jats:p>","DOI":"10.2307\/2586584","type":"journal-article","created":{"date-parts":[[2006,4,18]],"date-time":"2006-04-18T18:43:03Z","timestamp":1145385783000},"page":"29-49","source":"Crossref","is-referenced-by-count":0,"title":["Orthogonal families of real sequences"],"prefix":"10.1017","volume":"63","author":[{"given":"Arnold W.","family":"Miller","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Juris","family":"Steprans","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200015292_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(89)90013-4"},{"key":"S0022481200015292_ref002","first-page":"256","volume-title":"Set theory","author":"Kunen","year":"1980"},{"key":"S0022481200015292_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0016-660X(72)90001-3"},{"key":"S0022481200015292_ref004","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139172592"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200015292","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,11]],"date-time":"2019-05-11T19:03:51Z","timestamp":1557601431000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200015292\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,3]]},"references-count":4,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1998,3]]}},"alternative-id":["S0022481200015292"],"URL":"https:\/\/doi.org\/10.2307\/2586584","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,3]]}}}