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Topological classification

Finite dimensional Banach spaces are homeomorphic as topological spaces, if and only if they have the same dimension as real vector spaces.

Anderson–Kadec theorem (1965–66) proves that any two infinite-dimensional separable Banach spaces are homeomorphic as topological spaces. Kadec's theorem was extended by Torunczyk, who proved that any two Banach spaces are homeomorphic if and only if they have the same density character, the minimum cardinality of a dense subset.