Orthonormal basis for non-separable inner-product space

This is Problem 54 in Halmos' "A Hilbert Space Problem Book". However, I think this is a concrete counterexample. [Please let me know if not viewable.]

On the arXiv this (2010-09-09) morning:

1009.1441 [ps, pdf, other]

Title: Inner product space with no ortho-normal basis without choice.
Authors: Saharon Shelah
Primary Subject: math.LO

We prove in ZF that there is an inner product space, in fact, nicely definable with no orthonormal basis.

Sci.math, March 8, 2000 LINK