This is a well-known fact in Banach space theory (even locally convex space theory in a suitable form). See, for example, the classical monographs of Köthe or Schaefer. The direction you are looking for follows from the Alaoglu theorem (for the version involving compactness). For the variants, see the Eberlein-Smulian theorem, e.g., in Grothendieck's text.
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