Kadec–Klee property and fixed points

Let X be a reflexive Banach space which does not have the Kadec–Klee property. Then there exists a compact mapping f   from the unit ball BXBX of X   to the dual space X⁎X⁎ such that infx∈BX‖f(x)‖>0infx∈BX‖f(x)‖>0 and 〈f(x),x〉<‖f(x)‖〈f(x),x〉<‖f(x)‖ for every x∈BXx∈BX.