  We study uniform $\epsilon-$BPB approximations of bounded linear operators
between Banach spaces from a geometric perspective. We show that for
sufficiently small positive values of $\epsilon,$ many geometric properties
like smoothness, norm attainment and extremality of operators are preserved
under such approximations. We present examples of pairs of Banach spaces
satisfying non-trivial norm preserving uniform $\epsilon-$BPB approximation
property in the global sense. We also study these concepts in case of bounded
linear operators between Hilbert spaces. Our approach in the present article
leads to the improvement and generalization of some earlier results in this
context.
