Residuality in the set of norm attaining operators between Banach spaces

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