Definition of relative sequential compactness

1. Sep 15, 2014

DavideGenoa

Dear friends, I am used to the definition of relative sequential compactness of $X'$, a subset of a topological space $X$, as every sequence in $X'$ has a subsequence converging to a point in $X$.
I know that if $E$ is a separable Banach space then every weak$^{\ast}$-compact subset of $E^{\ast}$ is sequentially weak$^{\ast}$-compact and it seems to me that such an assertion can be generalised to "if $E$ is a separable Banach space then every relatively weak$^{\ast}$-compact subset of $E^{\ast}$ is relatively sequentially weak$^{\ast}$-compact".
Have I correctly desumed this generalisation?
I $\infty$-ly thank you for any help!

Last edited: Sep 15, 2014