Let $C$ be a convex subset of a Banach space $X$ and $T:C\to C$ a (norm) continuous affine map, i.e. $$T(tx+(1-t)y)=tT(x)+(1-t)T(y)$$ for $0\le t\le 1$. Is $T$ weakly continuous, i.e continuous as a map from $(C,\tau)$ to $(C,\tau)$ where $\tau$ is the topology induced by the weak topology of $X$.
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