Among Hausdorff locally convex spaces, semi-reflexivity is characterized by the weak topology having the Heine-Borel property. It follows that, in a semi-reflexive space, every nested net of closed bounded nonempty convex sets has nonempty intersection.
Stated in terms of sequence, this is actually a characterization of reflexivity in the normed case, by a theorem of Smulian. Moreover, Klee extended the result from nonempty convex sets to star-shaped sets. My two questions are whether those results extend to Hausdorff locally convex spaces:
- If a Hausdorff locally convex space satisfies that every nested net of closed bounded nonempty convex sets has nonempty intersection, is it semi-reflexive?
- If a Hausdorff locally convex space satisfies the hypothesis of 1, does it actually satisfy that every nested net of closed bounded star-shaped sets has nonempty intersection?
For references, see:
- Bernardes, Nilson C. jun., On nested sequences of convex sets in Banach spaces, J. Math. Anal. Appl. 389, No. 1, 558-561 (2012). ZBL1241.46007.
