For normed spaces with separable duals, weak-topology spaces are sequentially homeomorphic exactly when their classes of closed bounded weak subsets coincide; for hyperplane-isomorphic Banach spaces, the compact-convergence topologies are homeomorphic under the same condition.
Banakh, Toward a topological classification of convex sets in infinit e-dimensional Frchet spaces , Topology Appl
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On topological classification of normed spaces endowed with the weak topology or the topology of compact convergence
For normed spaces with separable duals, weak-topology spaces are sequentially homeomorphic exactly when their classes of closed bounded weak subsets coincide; for hyperplane-isomorphic Banach spaces, the compact-convergence topologies are homeomorphic under the same condition.