Abstract
G. Godefroy asked whether, on any Banach space, the set of norm-attaining functionals contains a 2-dimensional linear subspace.
We prove that a construction due to C.J. Read provides an example of a space which does not have this property.
Read found an equivalent norm (sic) center dot (sic) on c(0) such that (co, (sic) center dot (sic)) contains no proximinal subspaces of codimension 2. Our result is obtained through a study of the relation between the following two sentences, in which X is a Banach space and Y subset of X is a closed subspace: (A) Y is proximinal in X, and (B) Y-perpendicular to consists of norm-attaining functionals.
We prove that these are equivalent if X is the space (c(0), (sic) center dot (sic) )and our main theorem then follows as a corollary to Read's result.