Anderson–Kadec theorem

Remarks

Kadec proved that all infinite-dimensional separable Banach spaces are homeomorphic, and Anderson proved that $\R^{\N}$ is homeomorphic to $\ell^2$. The homeomorphisms are far from linear: $\ell^1$ and $\ell^2$ are homeomorphic but not isomorphic as Banach spaces. Both hypotheses are needed. In finite dimension, $\R^m$ and $\R^n$ are not homeomorphic for $m \ne n$, by invariance of domain, and neither is homeomorphic to $\R^{\N}$. Without separability it fails: $\ell^\infty$ is not separable, so it is not homeomorphic to $\ell^2$.

Going deeper

For a student

The result is the starting point of infinite-dimensional topology. Since all separable infinite-dimensional Fréchet spaces are homeomorphic, manifolds modelled on any of them can be compared purely topologically. For Hilbert manifolds the smooth counterpart is the open embedding theorem. It places every separable Hilbert manifold modelled on an infinite-dimensional separable Hilbert space inside its model space as an open set.