Weak compactness and weak sequential compactness in Banach spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T14:04:43Z http://mathoverflow.net/feeds/question/102103 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102103/weak-compactness-and-weak-sequential-compactness-in-banach-spaces Weak compactness and weak sequential compactness in Banach spaces Weeson Dorne 2012-07-13T04:15:44Z 2012-07-20T04:47:38Z <p>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 $$V=\lbrace x\in E;|(f_i,x-x_0)|\lt\epsilon, i\in I\rbrace$$<br> 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 $B(x_0,\delta)\subset V$. </p> http://mathoverflow.net/questions/102103/weak-compactness-and-weak-sequential-compactness-in-banach-spaces/102123#102123 Answer by Pietro Majer for Weak compactness and weak sequential compactness in Banach spaces Pietro Majer 2012-07-13T09:46:15Z 2012-07-16T05:53:04Z <p>Yes, it's the <a href="http://en.wikipedia.org/wiki/Eberlein%E2%80%93%C5%A0mulian_theorem" rel="nofollow">Eberlein–Šmulian theorem</a></p>