TITLE: Strict topology between weak and norm topologies
QUESTION [7 upvotes]: I want to believe that this has an easy answer, but I’ve never considered it before and can’t seem to answer it now either.

Does every infinite-dimensional Banach space admit a locally convex vector topology that is strictly coarser than the norm topology and strictly finer than the weak topology?

If this is non-trivial and constructive, I’d be much obliged if an explicit example (or reference) is provided.

REPLY [11 votes]: Another example is the topology of uniform convergence on the norm-compact subsets of the dual. It is coarser than the norm topology (which is the topology of uniform convergence on the dual unit ball) because compact sets are bounded and strictly coarser since the dual unit ball is not compact. It is finer than the weak topology since finite sets are compact and it is strictly finer because every linear independent null sequence in the dual together with its limit is compact and not finite dimensional.