  We completely characterize smoothness of bounded linear operators between
infinite dimensional real normed linear spaces, probably for the very first
time, by applying the concepts of Birkhoff-James orthogonality and
semi-inner-products in normed linear spaces. In this context, the key aspect of
our study is to consider norming sequences for a bounded linear operator,
instead of norm attaining elements. We also obtain a complete characterization
of smoothness of bounded linear operators between real normed linear spaces,
when the corresponding norm attainment set non-empty. This illustrates the
importance of the norm attainment set in the study of smoothness of bounded
linear operators. Finally, we prove that G\^{a}teaux differentiability and
Fr\'{e}chet differentiability are equivalent for compact operators in the space
of bounded linear operators between a reflexive Kadets-Klee Banach space and a
Fr\'{e}chet differentiable normed linear space, endowed with the usual operator
norm.
