Let $$u_m \rightharpoonup u$$ in $X$ (weak convergence). We are given $f:X \to X$, a continuous invertible map which is monotone $$(f(x)-f(y),x-y) \geq 0\quad\text{for all $x, y$}$$ and we have $$f(u_m) \rightharpoonup f(v)$$ in $X$ for some $v \in X$.

How do I show that indeed $v=u$, i.e. $f(u_m) \rightharpoonup f(u)$? I can't seem to do it by using the monotonicity method.