Are there known examples of compact infinite dimensional manifolds?
The word "manifold" is important.
Are there known examples of compact infinite dimensional manifolds? The word "manifold" is important. 


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The empty space is a manifold of any dimension. No, seriously, let's assume that "manifold" means a Hausdorff space in which every point has an open neighborhood homeomorphic to an open subset of a topological vector space. If the manifold is compact and nonempty then the vector space must be locally compact. As far as I know, that makes it finitedimensional. 


Yes. Compact Hilbert cube manifolds, for instance. 

