Proof of the topological equivalence of all separable infinite-dimensional Banach spaces

Quantitative geometry

“…Answering questions that were raised in the late 1920s and early 1930s (14,15), a deep theorem of Kadec (16) asserts that any two separable infinite dimensional Banach spaces are homeomorphic. On the other hand, Ribe proved (17) that any two Banach spaces X; Y that are uniformly homeomorphic (i.e., the homeomorphism and its inverse are both uniformly continuous) have the same isomorphic local linear structure, i.e., there exists K ∈ ð0; ∞Þ such that for any finite dimensional linear subspace E ⊆ X there exists a linear subspace F ⊆ Y that is linearly isomorphic to E via a linear mapping T : E → F satisfying jjTjj · jjT −1 jj ≤ K. Thus, while the rich world of separable infinite dimensional Banach spaces collapses to a single object when one considers them as topological spaces, a lot of structure remains if one insists that homeomorphisms are quantitatively continuous.…”

Extending certain transformation group actions in separable, infinite-dimensional Frechet spaces and the Hilbert cube

“…Since it has recently been shown ([l], [5] and [6]) that all separable, infinite-dimensional Frechet spaces are homeomorphic (a Frechet space being a metrizable, complete, locally convex linear topological space), this result holds in all such spaces. A salient property of compact sets in such spaces is that for each nonnull, homotopically trivial open set U, the relative complement in U of a compact set is also nonnull and homotopically trivial (see [3, § §3-5]).…”

Non-separable Hilbert manifolds of continuous mappings

“…From the topological viewpoint, it is well known that all separable Fréchet spaces (i.e., completely metrizable, locally convex real topological vector spaces) are mutually homeomorphic [1,4,8]. By the characterization result of Toruńczyk [14] (cf.…”