{"paper":{"title":"Orthogonal Families of Real Sequences","license":"","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Arnold W. Miller, Juris Stepr\\=ans","submitted_at":"1995-09-12T00:00:00Z","abstract_excerpt":"For x and y sequences of real numbers define the inner product (x,y) = x(0)y(0) + x(1)y(1)+ ... which may not be finite or even exist. We say that x and y are orthogonal iff (x,y) converges and equals 0.\n  Define l_p to be the set of all real sequences x such that |x(0)|^p + |x(1)|^p  + .. converges. For Hilbert space, l_2, any family of pairwise orthogonal sequences must be countable.\n Thm 1. There exists a pairwise orthogonal family F of size continuum such that F is a subset of l_p for every p>2.\n  It was already known that there exists a family of continuum many pairwise orthogonal element"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9509210","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}