Let $E$ be a Banach space and $T: E\rightarrow E$ be a mapping. $T$ is said to be

subcontinuousif for any sequences $(u_n)_{n\in\mathbb{N}}$ in $E$ that converge strongly to $u$ the sequence $(T(u_n))_{n\in\mathbb{N}}$ converges weakly to $T(u)$.

I am looking for a subcontinuous function which is not continuous. I thought I can find something in $l^2$, but I didn't.

*I already posted it on math.stackexchange, but to no avail*

Thank you