  For Banach spaces $X,Y,$ we consider a distance problem in the space of
bounded linear operators $\mathcal{L}(X,Y).$ Motivated by a recent paper
\cite{RAO21}, we obtain sufficient conditions so that for a compact operator
$T\in\mathcal{L}(X,Y)$ and a closed subspace $Z\subset Y,$ the following
equation holds, which relates global approximation with local approximation:
  \[d(T,\mathcal{L}(X,Z))=\sup\{d(Tx,Z):x\in X,\|x\|=1\}.\]
  In some cases, we show that the supremum is attained at an extreme point of
the corresponding unit ball. Furthermore, we obtain some situations when the
following equivalence holds:
  $$T\perp_B \mathcal{L}(X,Z)\Leftrightarrow T^{**}x_0^{**}\perp_B
Z^{\perp\perp}\Leftrightarrow
T^{**}\perp_B\mathcal{L}(X^{**},Z^{\perp\perp}),$$ for some $x_0^{**}\in
X^{**}$ satisfying $\|T^{**}x_0^{**}\|=\|T^{**}\|\|x_0^{**}\|,$ where $Z^\perp$
is the annihilator of $Z.$ One such situation is when $Z$ is an $L^1-$predual
space and an $M-$ideal in $Y$ and $T$ is a multi-smooth operator of finite
order. Another such situation is when $X$ is an abstract $L_1-$space and $T$ is
a multi-smooth operator of finite order. Finally, as a consequence of the
results, we obtain a sufficient condition for proximinality of a subspace $Z$
in $Y.$
