Does weak continuity imply continuity? I have come across the following excerpt from a mathematical Statistics book:

where $H$ and $J$ are Hilbert spaces and $H^{\star}$ is the dual space. For me, the statement after Definition 10 is unconvincing and I cannot, in general, show that weak continuity implies continuity. In particular, if $x_n$ converges strongly to $x$, then it also converges weakly and by weak continuity $f(x_n)$ converges weakly to $f(x)$. However, we need $f(x_n)$ to converge strongly to $f(x)$ in order to have continuity and it's not clear to me how we can obtain this.
I expect this to be true when $J$ is finite-dimensional as in that case weak and strong convergence are equivalent but other than that I am at a loss. I was wondering then, is the book wrong on this?

'Clearly' should not have been there but the result is true. This  require the so-called Closed Graph Theorem (CGT). If $x_n \to x$ and $f(x_n) \to  y$ in the norm then  $f(x_n) \to f(x)$ weakly and it converges to $y$ in the norm (hence also weakly) and this implies $y=f(x)$. By CGT it follows that $f$ is continuous.
