caa a22 4500 38803954X CHVBK 20180307125017.0 cr unu---uuuuu 161130e199803 xx s 000 0 eng 10.2307/2586584 doi S0022481200015292 pii (NATIONALLICENCE)cambridge-10.2307/2586584 Orthogonal families of real sequences [Elektronische Daten] For x, y ϵ ℝω define the inner product which may not be finite or even exist. We say that x and y are orthogonal if (x, y) converges and equals 0. Define lp to be the set of all x ϵ ℝω such that For Hilbert space, l2 , any family of pairwise orthogonal sequences must be countable. For a good introduction to Hilbert space, see Retherford [4]. Theorem 1. There exists a pairwise orthogonal family F of size continuum such that F is a subset of lp for every p > 2. It was already known that there exists a family of continuum many pairwise orthogonal elements of ℝω. A family F ⊆ ℝω∖0 of pairwise orthogonal sequences is orthogonally complete or a maximal orthogonal family iff the only element of ℝω orthogonal to every element of F is 0, the constant 0 sequence. 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. Theorem 2. There exists a perfect maximal orthogonal family of elements of ℝω . Abian raised the question of what are the possible cardinalities of maximal orthogonal families. Theorem 3. In the Cohen real model there is a maximal orthogonal set in ℝω of cardinality ω1, but there is no maximal orthogonal set of cardinality κ with ω1 < κ < ϲ. By the Cohen real model we mean any model obtained by forcing with finite partial functions from γ to 2, where the ground model satisfies GCH and γω = γ. Copyright © Association for Symbolic Logic 1998 Miller Arnold W. University of Wisconsin-Madison, Department of Mathematics, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706-1388, USA, E-mail: miller@math.wisc.edu, http://math.wisc.edu/~miller Steprans Juris York University, Department of Mathematics, North York, Ontario M3J 1P3, Canada, E-mail: juris.steprans@mathstat.yorku.ca The Journal of Symbolic Logic Cambridge University Press 63/1(1998-03), 29-49 0022-4812 63:1<29 1998 63 JSL https://doi.org/10.2307/2586584 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.2307/2586584 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Miller Arnold W. University of Wisconsin-Madison, Department of Mathematics, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706-1388, USA, E-mail: miller@math.wisc.edu, http://math.wisc.edu/~miller NATIONALLICENCE 700 1- Steprans Juris York University, Department of Mathematics, North York, Ontario M3J 1P3, Canada, E-mail: juris.steprans@mathstat.yorku.ca NATIONALLICENCE 773 0- The Journal of Symbolic Logic Cambridge University Press 63/1(1998-03), 29-49 0022-4812 63:1<29 1998 63 JSL CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge