Let $X$ be a normed space, and consider the weak-star topology for its dual. We know that the $w^*$continuous functionals are the evaluating ones, those are $\hat{x}: X^* \rightarrow \mathbb{K}$ defined by $\hat{x}(x^*) = x^*(x)$ for each $x \in X$. We consider know the mappings defined by $x^* \mapsto |\hat{x}(x^*)|$.
My question is: why the latter mappings are $w^*$-continuous? (I found the assertion of "$x^* \mapsto |\hat{x}(x^*)|$ are $w^*$-continuous since $\hat{x}$ are weak$^*$-continuous" in a book). Is it not supposed that the only $w^*$-continuous mappings are the evaluating ones?
I really appreciate an explanation of where is my reasoning wrong and why are the modulus of the evaluating mappings weak-star continuous, thank you!
The evaluation functionals $\hat{x}(x^*) = x^*(x)$ are linear continuous mappings from $X^*$ with the weak$^*$ topology to $\mathbb R$. If you compose that with the continuous map $t \mapsto |t|$, then the composition $x^* \mapsto |\hat{x}(x^*)|$ is a continuous (nonlinear) function from $X^*$ with the weak$^*$ topology to $\mathbb R$.