# Density of norm-attaining operators

By Bishop-Phelps theorem we know that for a real Banach space, the set of all norm attaining bounded linear functionals is norm-dense in $$X^*$$, the topological dual of $$X$$. We also know that in general the class of all norm attaining operators $$NA(X,Y)$$ from a Banach space $$X$$ into another Banach space $$Y$$ need not be dense in the space of all bounded linear operators $$\mathcal{B}(X,Y)$$. I am interested in one particular example where $$NA(X,Y)$$ is not dense in $$\mathcal{B}(X,Y)$$.

In his paper titled "On operators which attain their norm", Lindenstrauss mentioned that the simplest of such examples can be obtained by using the fact that a one-one norm-attaining operator from a Banach space to a strictly convex space attains its norm at an extreme point of $$B_X$$, the unit closed ball of $$X$$. But I could not make it out completely. Can anyone please suggest me how the extreme point help one to get the desired example?

Take a separable $$X$$ s.t. $$B_X$$ has no extreme points (for example, $$c_0$$ or $$L_1$$), and equivalently renorm it to be strictly convex--call the resulting space $$Y$$. Show that the identity operator from $$X$$ to $$Y$$ cannot be approximated by norm attaining operators.