TAMENESS AND ROSENTHAL TYPE LOCALLY CONVEX SPACES

The paper’s Theorem 10.13 is proved with a precise chain of implications ((1)⇒(2), (2)⇒(3), (3)⇒(4), (4)⇒(1), and (2)⇔(5)), relying on earlier lemmas and propositions in the text. In contrast, the candidate solution reverses a key direction, asserting (4)⇒(3) at the M-level and (iv)⇒(iii) at the global level by applying the Haydon property directly to sets of extreme points. This is not justified because ext N (or ext C) need not be weak-* closed/compact, so the property co^{w*}(L)=co(L) for weak-* closed L cannot be invoked as done. The paper avoids this pitfall by proving (3)⇒(4) and using Lemma 10.11 for (4)⇒(1). Aside from these directional errors, several other steps in the model’s outline are either underspecified or invoke external results without the glue provided in the paper.