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If $E$ is a Banach space, $A$ is a subset of $E$ and is compact with the weak topology $\sigma(E,E')$, that is the most coarse topology which make every $f\in E'$ continuous, is it true that $A$ is weakly sequentially compact? As we all know, in $C_1$ spaces, compact concludes sequentially compact. So, we should show that $E$ is a $C_1$ spaces with the topology $\sigma(E,E')$. Some known conclusions:$\forall x_0\in E$ , the basis of neighborhoods of $x_0$ constitutes of $V$ with the below form $$V=\lbrace x\in E;|(f_i,x-x_0)|\lt\epsilon, i\in I\rbrace$$
where I is a finite set. Can we prove $E$ is a $C_1$ spaces? Now, for every such $V$, we can find a $\delta>0$, such that $B(x_0,\delta)\subset V$.

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weakcompactWeakcompactness and weak selfsequencecompactsequentialcompactnessinBanachspaces

If $E$ is a Banach space, $A$ is a subset of $E$ and is compact with the weak topology $\sigma(E,E')$, that is the most coarse topology which make every $f\in E'$ continuous, is it true that $A$ is weakly sequentially compact? As we all know, in $C_1$ spaces, compact concludes sequentially compact. So, we should show that $E$ is a $C_1$ spaces with the topology $\sigma(E,E')$. Some known conclusions:$\forall x_0\in E$ , the basis of neighborhoods of $x_0$ constitutes of $V$ with the below form $$V={x\in E;|(f_i,x-x_0)|<\epsilon, V=\lbrace x\in E;|(f_i,x-x_0)|\lt\epsilon, i\in I}$$I\rbrace$$where I is a finite set.\ set. Can we prove E is a C_1 spaces? 5 detailed description of the topology; [made Community Wiki] If E is a Banach space, A is a subset of E and is compact with the weak topology \sigma(E,E'), that is the most coarse topology which make every f\in E' continuous, is it true that A is weakly sequentially compact? As we all know, in C_1 spaces, compact concludes sequentially compact. So, we should show that E is a C_1 spaces with the topology \sigma(E,E'). Some known conclusions:\forall x_0\in E , the basis of neighborhoods of x_0 constitutes of V with the below form$$V={x\in E;|(f_i,x-x_0)|<\epsilon, i\in I}
where I is a finite set.\ Can we prove $E$ is a $C_1$ spaces?

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