Un contre-exemple pour un espace d'interpolation qui n'est pas faiblement LUR

Convexity in the interpolation spaces

“…Remak 7.3. In [6] we show that, there exists an interpolation couple (B 0 , B 1 ) such that B 0 is a subspace of B 1 , B * 0 admits an equivalent LUR norm (hence this norm is rotund) but B * θ , B * θ,p not admit any equivalent rotund norms. Thus the condition that the dual norm of B * 0 is rotund is necessary.…”

“…In [5], we show that if the norm on B 0 is W UR, then the norm on B θ is W UR. We recall by [6], there exists an interpolation couple (A 0 , A 1 ) such that the norm on A 0 is LUR but the interpolated spaces A θ , A θ,p not admit any equivalent rotund norm for all 0 < θ < 1 and all 1 < p < +∞.…”