Editing Banach space (section)

====Homeomorphism classes of separable Banach spaces====

All finite–dimensional normed spaces are separable Banach spaces and any two Banach spaces of the same finite dimension are linearly homeomorphic. 
Every separable infinite–dimensional [[Hilbert space]] is linearly isometrically isomorphic to the separable Hilbert [[ℓ2 space|sequence space <math>\ell^2(\N)</math>]] with its usual norm <math>\|{\cdot}\|_2.</math> 

The [[Anderson–Kadec theorem]] states that every infinite–dimensional separable [[Fréchet space]] is [[Homeomorphism|homeomorphic]] to the [[product space]] <math display=inline>\prod_{i \in \N} \Reals</math> of countably many copies of <math>\Reals</math> (this homeomorphism need not be a [[linear map]]).<ref>{{harvnb|Bessaga|Pełczyński|1975|p=189}}</ref>{{sfn|Anderson|Schori|1969|p=315}} 
Thus all infinite–dimensional separable Fréchet spaces are homeomorphic to each other (or said differently, their topology is unique [[up to]] a homeomorphism). <!-- and so as with finite–dimensional spaces, any two separable Fréchet spaces (of any dimensions) are homeomorphic if and only if they have the same dimension.<ref group=note>This means that their dimensions are either both finite and equal or else both infinite.</ref>-->
Since every Banach space is a Fréchet space, this is also true of all infinite–dimensional separable Banach spaces, including <math>\ell^2(\N).</math> 
In fact, <math>\ell^2(\N)</math> is even [[Homeomorphism|homeomorphic]] to its own [[Unit sphere|unit {{em|sphere}}]] <math>\{x \in \ell^2(\N) \mid \|x\|_2 = 1\},</math> which stands in sharp contrast to finite–dimensional spaces (the [[Euclidean plane]] <math>\Reals^2</math> is not homeomorphic to the [[unit circle]], for instance). 

This pattern in [[homeomorphism class]]es extends to generalizations of [[Metrizable topological space|metrizable]] ([[locally Euclidean]]) [[topological manifold]]s known as {{em|metric [[Banach manifold]]s}}, which are [[metric space]]s that are around every point, [[locally homeomorphic]] to some open subset of a given Banach space (metric [[Hilbert manifold]]s and metric [[Fréchet manifold]]s are defined similarly).{{sfn|Anderson|Schori|1969|p=315}} 
For example, every open subset <math>U</math> of a Banach space <math>X</math> is canonically a metric Banach manifold modeled on <math>X</math> since the [[inclusion map]] <math>U \to X</math> is an [[Open map|open]] [[local homeomorphism]]. 
Using Hilbert space [[microbundle]]s, David Henderson showed{{sfn|Henderson|1969|p=}} in 1969 that every metric manifold modeled on a separable infinite–dimensional Banach (or [[Fréchet space|Fréchet]]) space can be [[Topological embedding|topologically embedded]] as an [[Open set|{{em|open}} subset]] of <math>\ell^2(\N)</math> and, consequently, also admits a unique [[smooth structure]] making it into a <math>C^\infty</math> [[Hilbert manifold]].