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Definition of relative sequential compactness

  1. Sep 15, 2014 #1
    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
  2. jcsd
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