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I'm reading some papers where the condition of weak sequential continuity is crucial instead of the weak continuity.

So, let

• $T$ an operator between a Banach space $X$ and itself.

1. $T$ is weakly sequentially continuous if for any $(x_n)_{n\in \mathbb N}$, $x_n \stackrel{w}{\rightharpoonup}x$ in $X$ $\Rightarrow$ $T(x_n) \stackrel{w}{\rightharpoonup } T( x )$ in $X$)*.
2. $T$ is weakly continuous iff $T$ is continuous with respect to the weak topologies on $X$

While it is clear that 2 implies 1, I want to find a counter-example of an operator which verifies 1 but not 2.

I'm reading some papers where the condition of weak sequential continuity is crucial instead of the weak continuity.

So, let

• $T$ an operator between a Banach space $X$ and itself.

1. $T$ is weakly sequentially continuous if for any $(x_n)_{n\in \mathbb N}$, $x_n \stackrel{w}{\rightharpoonup}x$ in $X$ $\Rightarrow$ $T(x_n) \stackrel{w}{\rightharpoonup } T( x )$ in $X$)*.
2. $T$ is weakly continuous iff $T$ is continuous with respect to the weak topologies on $X$

While it is clear that 2 implies 1, I want to find a counter-example of an operator which verifies 1 but not 2.

$T$ is not linear

Im reading some papers where the condition of weak sequential continuity is crucial instead of the weak continuity.

So, let

• $T$ an operator between a Banach space $X$ and itself.

1. $T$ is weakly sequentially continuous if for any $(x_n)_{n\in \mathbb N}$, $x_n \stackrel{w}{\rightharpoonup}x$ in $X$ $\Rightarrow$ $T(x_n) \stackrel{w}{\rightharpoonup } T( x )$ in $X$)*.
2. $T$ is weakly continuous iff $T$ is continuous with respect to the weak topologies on $X$

While it is clear that 2 implies 1, I want to find a counter-example of an operator which verifies 1 but not 2.

I'm reading some papers where the condition of weak sequential continuity is crucial instead of the weak continuity.

So, let

• $T$ an operator between a Banach space $X$ and itself.

1. $T$ is weakly sequentially continuous if for any $(x_n)_{n\in \mathbb N}$, $x_n \stackrel{w}{\rightharpoonup}x$ in $X$ $\Rightarrow$ $T(x_n) \stackrel{w}{\rightharpoonup } T( x )$ in $X$)*.
2. $T$ is weakly continuous iff $T$ is continuous with respect to the weak topologies on $X$

While it is clear that 2 implies 1, I want to find a counter-example of an operator which verifies 1 but not 2.