  G. Godefroy asked whether, on any Banach space, the set of norm-attaining
functionals contains a 2-dimensional linear subspace. We prove that a recent
construction due to C.J. Read provides an example of a space which does not
have this property. This is done through a study of the relation between the
following two sentences where X is a Banach space and Y is a closed subspace of
finite codimension in X: (A) Y is proximinal in X. (B) The annihilator of Y
consists of norm-attaining functionals. We prove that these are equivalent if X
is the Read's space. Moreover, we prove that any non-reflexive Banach space X
with any given closed subspace Y of finite codimension at least 2 admits an
equivalent norm such that (B) is true and (A) is false.
