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Let $A$ be a convex and weakly compact subset of a Banach space $X$, and $\{x_n\}$ is a sequence in $A$, can we say $\{x_n\}$ has a weakly convergence sub sequence in $A$ ?

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  • $\begingroup$ Isn't that just the definition of weak compactness? $\endgroup$ – Roberto Rastapopoulos Mar 12 '17 at 0:52
  • $\begingroup$ No, notice the word "sub sequence". Weak topology is not magnetizable in general. $\endgroup$ – Red shoes Mar 12 '17 at 0:53
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I found the Answer of my own question :)

Thanks to Eberlein_ Smulian theorem the answer is positive!

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