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It is known that, if a function $f$ from a planar domain $D$ to a Banach space $A$ is weakly analytic [i.e. $l(f)$ is analytic for every $l$ in $A^*$], then $f$ is strongly analytic [i.e. $\lim_{h \to 0} h^{-1}[f(z+h)-f(z)]$ exists in norm for every $z$ in $D$].

Now the question is, if above $f$ is assumed to be weakly continuous [i.e.$l(f)$ is continuous for every $l$ in $A^*$], then is it true that $f$ will be strongly continuous.[i.e. $\lim_{h \to 0} [f(z+h)-f(z)] = 0$ in norm for every $z$ in $D$.]

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    It is well known that weakly analytic function on a planar domain is strongly analytic.But what about continuous function?2012-09-20

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Regarding the clarified question with finite dimensional domain. Let $X$ be an infinite dimensional separable and reflexive Banach space. Its unit ball $B$ is weakly compact and metrizable. It is also convex.

So by Hahn–Mazurkiewicz theorem there exists a continuous function $f\colon [0,1]\to B$ that is onto $B$. This function is not norm continuous as $B$ is not norm compact.

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    there is definitely no function in explicit form. Basically, one shows that the set $B$ is the image of the cantor set. try finding an explicit form for a path filling curve from $[0,1]$ onto $[0,1]^2$. Now try finding one for the unit ball of $L_2$. I'll add, the answer to your question is that weak continuity does not imply strong continuity even if the domain is finite dimensional; and we can find such an example for every metrizable domain in which we embed $[0,1]$ as a closed set.2012-09-20
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Let $X$ be an infinite dimensional Banach space. Let $T_w$ denote the weak topology and let $T$ denote the norm topology. Then $\mathrm{id}: (X, T_w) \to (X, T)$ is not continuous but $\mathrm{id}: (X, T_w) \to (X, T_w)$ is.

Note that every map $f: X \to Y$ that is weakly continuous where weakly means $f: (X, T_w) \to Y$ is also strongly continuous, $f: (X, T) \to Y$, since the norm (or strong) topology contains more open sets than the weak topology (by definition).

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    @timon Nice, thanks!2012-09-20