Orthogonal Families of Real Sequences

Abstract

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. 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. 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. It was already known that there exists a family of continuum many pairwise orthogonal elements of real sequences. Thm 2. There exists a perfect maximal orthogonal family of elements of real sequences. Abian raised the question of what are the possible cardinalities of maximal orthogonal families. Thm 3. In the Cohen real model there is a maximal orthogonal set cardinality omega_1, but there is no maximal orthogonal set of cardinality k with \omega_1< k < c. Thm 4. For any countable standard model M of ZFC and cardinal k in M such that M satisfies k^\omega=k, there exists a ccc generic extension M[G] such that the continuum of M[G] is k and in M[G] for every infinite cardinal i less than or equal to k there is a maximal orthogonal family of cardinality i. Thm 5. (MA_k(\sigma-centered)) Suppose cardinality of X is less than or equal to k, X contains only finitely many elements of l_2, and for every distinct pair x,y in X the inner product (x,y) converges. Then there exists a z such that z is orthogonal to every element of X. Thm 6.(a) There exists X which is a maximal orthogonal family in l_2 such that for all n with 1\leq n \leq\omega there exists Y of cardinality n with (X union Y) a maximal orthogonal family in $\rr^\om$. Furthermore, every maximal orthogonal family containing $X$ is countable. (b) There exists a perfect maximal orthogonal family P such that (P intersect l_2) is a maximal orthogonal family in $l_2$.