  We show that if the conclusion of the well known Stampacchia Theorem, on
variational inequalities, holds on a Banach space X, then X is isomorphic to a
Hilbert space. Motivated by this we obtain a relevant result concerning
self-dual Banach spaces and investigate some connections between existing
notions of orthogonality and self-duality. Moreover, we revisit the notion of
the cosine of a linear operator and show that it can be used to characterize
Hilbert space structure. Finally, we present some consequences of our results
to quadratic forms and to evolution triples.
