We study the Bishop-Phelps-Bollobàs property
$\left( \text{BPBp} \right)$
for compact operators. We present some abstract techniques that allow us to carry the
$\text{BPBp}$
for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let
$X$
and
$Y$
be Banach spaces. If
$\left( {{c}_{0}},Y \right)$
has the
$\text{BPBp}$
for compact operators, then so do
$\left( {{C}_{0}}\left( L \right),Y \right)$
for every locally compact Hausdorff topological space
$L$
and
$\left( X,\,Y \right)$
whenever
${{X}^{*}}$
is isometrically isomorphic to
${{\ell }_{1}}$
.
If
${{X}^{*}}$
has the Radon-Nikodým property and
$\left( {{\ell }_{1}}\left( X \right),\,Y \right)$
has the
$\text{BPBp}$
for compact operators, then so does
$\left( {{L}_{1}}\left( \mu ,X \right),\,\,Y \right)$
for every positive measure
$\mu $
; as a consequence,
$\left( {{L}_{1}}\left( \mu ,X \right),\,\,Y \right)$
has the
$\text{BPBp}$
for compact operators when
$X$
and
$Y$
are finite-dimensional or
$Y$
is a Hilbert space and
$X={{c}_{0}}$
or
$X={{L}_{p}}\left( v \right)$
for any positive measure
$v$
and
$1\,<\,p\,<\,\infty $
. For
$1\,\le p\,<\,\infty$
, if
$\left( X,{{l}_{p}}(Y) \right)$
has the
$\text{BPBp}$
for compact operators, then so does
$\left( X,{{L}_{p}}\left( \mu ,\,Y \right) \right)$
for every positive measure
$\mu $
such that
${{L}_{1}}\left( \mu \right)$
is infinite-dimensional. If
$\left( X,\,Y \right)$
has the
$\text{BPBp}$
for compact operators, then so do
$\left( X,\,{{L}_{\infty }}\left( \mu ,\,\,Y \right) \right)$
for every
$\sigma $
-finite positive measure
$\mu $
and
$\left( X,\,C\left( K,\,Y \right) \right)$
for every compact Hausdorff topological space
$K$
.