in Fixed point theory, Let X be a normed space and $\mu$ a gauge function. Then the mapping $ J_{\mu}:X\longrightarrow {2^{X^*}}$ defined by
$$J_{\mu}(x)=\{j\in X^*: (x,j)=\|x\|\|j\|_*, \|j\|_*=(\mu(\|x\|)\}$$
is called the duality mapping with gauge function $\mu$.
Now, Let X be a reflexive Banach space and $J^*:X^*\longrightarrow X$ the inverse of the normalized duality mapping $J:X\longrightarrow X^*$ Then $$J^*J=I , JJ^*=I$$ where $$J^*(j)=J^{-1}(j)=\{x\in X: (x,j)=\|x\|\|j\|_*, \|x\|=(\mu^{-1}(\|j\|_*)\}$$