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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?

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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.

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