  In this paper we completely characterize the norm attainment set of a bounded
linear operator on a Hilbert space. This partially answers a question raised
recently in [\textit{D. Sain, On the norm attainment set of a bounded linear
operator, accepted in Journal of Mathematical Analysis and Applications}]. We
further study the extreme contractions on various types of finite-dimensional
Banach spaces, namely Euclidean spaces, and strictly convex spaces. In
particular, we give an elementary alternative proof of the well-known
characterization of extreme contractions on a Hilbert space, that works equally
well for both the real and the complex case. As an application of our
exploration, we prove that it is possible to characterize real Hilbert spaces
among real Banach spaces, in terms of extreme contractions on two-dimensional
subspaces of it.
