Anderson–Kadec theorem
In mathematics, in the areas of topology and functional analysis, the Anderson–Kadec theorem states[1] that any two infinite-dimensional, separable Banach spaces, or, more generally, Fréchet spaces, are homeomorphic as topological spaces.The theorem was proved by Mikhail Kadec (1966) and Richard Davis Anderson.Every infinite-dimensional, separable Fréchet space is homeomorphic tothe Cartesian product of countably many copies of the real lineis called a Kadec norm with respect to a total subsetthe following condition is satisfied: Eidelheit theorem: A Fréchet spaceKadec renorming theorem: Every separable Banach spaceadmits a Kadec norm with respect to a countable total subsetcan be taken to be any weak-star dense countable subset of the unit ball ofdenotes an infinite-dimensional separable Fréchet space andthe relation of topological equivalence (existence of homeomorphism).A starting point of the proof of the Anderson–Kadec theorem is Kadec's proof that any infinite-dimensional separable Banach space is homeomorphic toA result of Bartle-Graves-Michael proves that thenis a closed subspace of a countable infinite product of separable Banach spacesThe same result of Bartle-Graves-Michael applied toFrom Kadec's result the countable product of infinite-dimensional separable Banach spacesThe proof of Anderson–Kadec theorem consists of the sequence of equivalences