  We study the relationship between the residuality of the set of norm
attaining functionals on a Banach space and the residuality and the denseness
of the set of norm attaining operators between Banach spaces. Our first main
result says that if $C$ is a bounded subset of a Banach space $X$ which admit
an LUR renorming satisfying that, for every Banach space $Y$, the operators $T$
from $X$ to $Y$ for which the supremum of $\|Tx\|$ with $x\in C$ is attained
are dense, then the $G_\delta$ set of those functionals which strongly exposes
$C$ is dense in $X^*$. This extends previous results by J.\ Bourgain and K.-S.\
Lau. The particular case in which $C$ is the unit ball of $X$, in which we get
that the norm of $X^*$ is Fr\'{e}chet differentiable at a dense subset,
improves a result by J.\ Lindenstrauss and we even present an example showing
that Lindenstrauss' result was not optimal. In the reverse direction, we obtain
results for the density of the $G_\delta$ set of absolutely strongly exposing
operators from $X$ to $Y$ by requiring that the set of strongly exposing
functionals on $X$ is dense and conditions on $Y$ or $Y^*$ involving RNP and
discreteness on the set of strongly exposed points of $Y$ or $Y^*$. These
results include examples in which even the denseness of norm attaining
operators was unknown. We also show that the residuality of the set of norm
attaining operators implies the denseness of the set of absolutely strongly
exposing operators provided the domain space and the dual of the range space
are separable, extending a recent result for functionals. Finally, our results
find important applications, among which we point out that we solve a proposed
open problem showing that the unique predual of the space of Lipschitz
functions from the Euclidean unit circle fails to have Lindenstrauss property
A.
